# Generic coarse geometry of leaves

Jesús A. Álvarez López, Alberto Candel

► To cite this version:

Jesús A. Álvarez López, Alberto Candel. Generic coarse geometry of leaves. Springer International Publishing; Lecture Notes in Mathematics, 2223, 2018, Lecture Notes in Mathematics, 978-3-319-94132-5. 10.1007/978-3-319-94132-5_1. hal-04410880

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Distributed under a Creative Commons Attribution 4.0 International License
Generic Geometry of Leaves

Jesús A. Álvarez López<fnref n="1" />

Departamento de Xeometría e Topoloxía
Facultade de Matemáticas

Universidade de Santiago de Compostela
Campus Universitario Sur

15706 Santiago de Compostela, Spain

Alberto Candel<fnref n="2" />

Department of Mathematics
CSUN

Northridge, CA 91330
U.S.A.

November 1999
Revised ??? 2003

<footnote n="1">¹Research of first author supported by DGICYT Grant PB95-0850.</footnote>

<footnote n="2">²Research of second author supported by NSF Grant No. DMS-9973086 and DMS-0049077.</footnote>
# Contents

<table><tr><td>1</td><td>Introduction</td><td>1</td></tr><tr><td>2</td><td>Foliated Spaces and Pseudogroups</td><td>5</td></tr><tr><td>2.1</td><td>Topological Preliminaries</td><td>5</td></tr><tr><td>2.2</td><td>Foliated spaces</td><td>6</td></tr><tr><td>2.3</td><td>Pseudogroups of local transformations</td><td>15</td></tr><tr><td>3</td><td>Metric structures on leaves</td><td>19</td></tr><tr><td>3.1</td><td>Coarse quasi-isometries</td><td>19</td></tr><tr><td>3.2</td><td>Coarse structures</td><td>26</td></tr><tr><td>3.3</td><td>Metric equivalence relations and length structures</td><td>35</td></tr><tr><td>3.4</td><td>Coarse quasi-isometry type of orbits</td><td>37</td></tr><tr><td>3.5</td><td>Fölner orbits and invariant measures</td><td>45</td></tr><tr><td>3.6</td><td>An example of non-recurrent compact generation</td><td>48</td></tr><tr><td>3.7</td><td>Good metrics on the leaves</td><td>50</td></tr><tr><td>3.8</td><td>Good length structures</td><td>56</td></tr><tr><td>3.9</td><td>Coarse quasi-isometry type of leaves</td><td>56</td></tr><tr><td>4</td><td>Topological Dynamics</td><td>61</td></tr><tr><td>4.1</td><td>Topological dynamics</td><td>61</td></tr><tr><td>4.2</td><td>A discrete version of local Reeb stability</td><td>68</td></tr><tr><td>5</td><td>Generic coarse geometry of leaves</td><td>71</td></tr><tr><td>5.1</td><td>Coarsely quasi-isometric leaves</td><td>71</td></tr><tr><td>5.2</td><td>Ends of coarsely quasi-symmetric leaves</td><td>76</td></tr><tr><td>5.3</td><td>Growth of leaves</td><td>77</td></tr><tr><td>5.4</td><td>Amenable leaves</td><td>84</td></tr><tr><td>6</td><td>Compacifications</td><td>89</td></tr><tr><td>6.1</td><td>Generalities on Compacifications</td><td>89</td></tr><tr><td>6.2</td><td>The Higson Compactification</td><td>90</td></tr><tr><td>6.3</td><td>The Higson Compactification of a Metric Space</td><td>95</td></tr><tr><td>6.4</td><td>Compacifications and Limit Sets</td><td>101</td></tr></table>
<table>
  <tbody>
    <tr>
      <td>6.5</td>
      <td>Asymptotic dimension of leaves</td>
      <td>103</td>
    </tr>
    <tr>
      <td><b>7</b></td>
      <td><b>Generic cohomology</b></td>
      <td><b>107</b></td>
    </tr>
    <tr>
      <td>7.1</td>
      <td>Coarse cohomology of the leaves</td>
      <td>107</td>
    </tr>
    <tr>
      <td>7.2</td>
      <td>Uniformly finite homology of the leaves</td>
      <td>108</td>
    </tr>
    <tr>
      <td>7.3</td>
      <td>Algebraic asymptotic invariants</td>
      <td>108</td>
    </tr>
    <tr>
      <td><b>8</b></td>
      <td><b>Generic differentiable geometry of leaves</b></td>
      <td><b>111</b></td>
    </tr>
    <tr>
      <td>8.1</td>
      <td>Differentiable quasi-isometries</td>
      <td>111</td>
    </tr>
    <tr>
      <td>8.2</td>
      <td>Quasi-isometric leaves</td>
      <td>115</td>
    </tr>
    <tr>
      <td><b>9</b></td>
      <td><b>A canonical Riemannian foliated space</b></td>
      <td><b>121</b></td>
    </tr>
    <tr>
      <td>9.1</td>
      <td>The Gromov space</td>
      <td>121</td>
    </tr>
    <tr>
      <td>9.2</td>
      <td>Convergence of Riemannian manifolds</td>
      <td>124</td>
    </tr>
    <tr>
      <td>9.3</td>
      <td>Compact families of Riemannian manifolds</td>
      <td>127</td>
    </tr>
    <tr>
      <td>9.4</td>
      <td>A canonical Riemannian foliated space</td>
      <td>132</td>
    </tr>
    <tr>
      <td><b>10</b></td>
      <td><b>Distortions and bidistortions</b></td>
      <td><b>149</b></td>
    </tr>
    <tr>
      <td>10.1</td>
      <td>Preliminaries and notation</td>
      <td>149</td>
    </tr>
    <tr>
      <td>10.2</td>
      <td>Underlying functors</td>
      <td>157</td>
    </tr>
    <tr>
      <td>10.3</td>
      <td>Local categories</td>
      <td>166</td>
    </tr>
    <tr>
      <td>10.4</td>
      <td>Distortions</td>
      <td>169</td>
    </tr>
    <tr>
      <td>10.5</td>
      <td>Examples of distortions</td>
      <td>172</td>
    </tr>
    <tr>
      <td>10.6</td>
      <td>Crossings</td>
      <td>179</td>
    </tr>
    <tr>
      <td>10.7</td>
      <td>Bidistortions</td>
      <td>181</td>
    </tr>
    <tr>
      <td>10.8</td>
      <td>A-functors and a-natural transfomations</td>
      <td>186</td>
    </tr>
    <tr>
      <td>10.9</td>
      <td>The completion defined by a bidistortion</td>
      <td>189</td>
    </tr>
    <tr>
      <td>10.10</td>
      <td>Pseudo-metrics defined by a bidistortion</td>
      <td>197</td>
    </tr>
    <tr>
      <td>10.11</td>
      <td>The Hausdorff property</td>
      <td>206</td>
    </tr>
    <tr>
      <td>10.12</td>
      <td>Examples of bidistortions</td>
      <td>211</td>
    </tr>
    <tr>
      <td>10.13</td>
      <td>Partitioned sets and spaces</td>
      <td>217</td>
    </tr>
    <tr>
      <td><b>11</b></td>
      <td><b>Universal foliated spaces</b></td>
      <td><b>221</b></td>
    </tr>
    <tr>
      <td>11.1</td>
      <td>A universal foliated space</td>
      <td>221</td>
    </tr>
    <tr>
      <td>11.2</td>
      <td>Generalized Gromov space</td>
      <td>239</td>
    </tr>
    <tr>
      <td>11.3</td>
      <td>Ideas of the proofs</td>
      <td>240</td>
    </tr>
    <tr>
      <td><b>12</b></td>
      <td><b>Dynamics in Gromov spaces</b></td>
      <td><b>241</b></td>
    </tr>
    <tr>
      <td>12.1</td>
      <td>Compactness of classes of metric spaces</td>
      <td>241</td>
    </tr>
    <tr>
      <td>12.2</td>
      <td>Dynamics</td>
      <td>244</td>
    </tr>
    <tr>
      <td>12.3</td>
      <td>Asymptotic cones</td>
      <td>247</td>
    </tr>
    <tr>
      <td>12.4</td>
      <td>The global Gromov space and equicontinuous foliated spaces</td>
      <td>252</td>
    </tr>
  </tbody>
</table>
# Chapter 1

# Introduction

The theme of the present study is the generic geometry of the orbits of a dynamical system, specially the leaves of a foliated space. A leaf of a compact foliated space has a well defined quasi-isometry type, which from a coarse point of view is that of the corresponding orbit of the holonomy pseudogroup. The question: “What do the leaves of a foliated space look like?” has been studied numerous times, and from this point of view, it has the following concise answer.

**Theorem.** A metric space is coarse quasi-isometric to a leaf of a compact foliated space if and only if it is coarse quasi-isometric to a homogeneous space $G/H$ of a finitely generated group $G$.

This is based on an elementary construction, and while it does not answer the question in a satisfactory way (as if there was a satisfactory answer), it does provide a technical tool for other arguments.

The question also arises as to how many different quasi-isometry types can occur among the leaves of a single compact lamination. As the way of relating different leaves must have something to do with recurrence, a natural condition to impose on the foliated space is that it be minimal (or transitive). A study of the relation “being quasi-isometric” among the leaves of a foliated space, shows that its equivalence classes are baire sets, and so basic topological dynamics provides the following answer.

**Theorem.** In a foliated space, either there is a residual set all whose leaves are quasi-isometric, or else every family of mutually quasi-isometric leaves is meager.

A further study of the equivalence relation sheds more light on the problem. It happens that the second possibility of the previous statement is rather restrictive. For the purpose of this introduction (see Definition ??), a leaf is quasi-symmetric if it has sufficiently many quasi-isometries of a fixed bounded distortion.

**Theorem.** There is a residual set of quasi-isometric leaves if and only if there is a residual set of quasi-symmetric leaves.
Other conditions implying almost all the leaves are quasi-isometric can be found ****

From a different point of view, it is also natural to find dynamical properties of a foliated
space implying that all the leaves are quasi-isometric. The example that comes to mind
is the case of Riemannian foliations, for it follows from Molino’s theory that the holonomy
covers of all the leaves are quasi-isometric via diffeomorphism. The topological analog of a
Riemannian foliation is a foliated space whose holonomy pseudogroup is equicontinuous. An
analysis of its structure shows that it has properties similar to those of a group of isometries.
This is the topic of the paper [1].

**Theorem.** The holonomy covers of all the leaves of topologically analytic equicontinuous foliated space are quasi-isometric.

On a slightly different direction, another natural question to ask is what kind of asymptotic
invariants of metric spaces are generic for the leaves of a foliated space, *i.e.* the same on a
residual set. Perhaps the best known asymptotic invariant in foliation theory is the order of
growth of a leaf.

**Theorem.**

Other well-known invariant is the number of ends. Here the model result is a theorem
of E. Ghys (in a measure theoretic context) and J. Cantwell and L. Conlon (in the topolog-
ical context taken here) which says that (under suitable ergodic or minimal hypothesis) the
number of ends of the leaves is essentially constant, and moreover, almost all the leaves have
either 0, 1, 2 or a Cantor set of ends.

The number of ends being a quasi-isometry invariant, this result of Ghys and Cantwell-
Conlon suggests the possibility of studying whether other asymptotic invariants of leaves are
generic. A large number of quasi-isometry invariants of metric spaces amenable to study fit
into homotopy functors. Let $F$ be a functor from the category of metric spaces to a category
with limits $\Lambda$. If $F$ is continuous (in a reasonable sense), then an asymptotic invariant of a
space $X$ can be defined as $F^\infty(X) = \lim_K F(X \setminus K)$, where $K$ runs over all compact subsets
$K$ of $X$.

**Theorem.** Let $F$ be a continuous functor with values in the category of vector spaces. Then $F^\infty(L)$ are isomorphic for a residual set of leaves $L$ of a given foliated space.

Other type of invariants are played by compactifications of the leaves. A relevant com-
pactification of a leaf, from the point of view of carrying quasi-isometric information, is that
of Higson-Roe. Its study has two aspects, one that relates to recurrence, and studies the
limit sets of points in the Higson corona.

**Theorem.** The limit sets of every point of the Higson corona is a union of leaves. With some minimality assumptions, the points of the Higson corona are recurrent.

On the side of asymptotic invariants, there is one related to the Higson compactification,
namely, the asymptotic dimension, as it is shown in []. This is an invariant that gives a rough
idea of the size of the Higson corona of a proper metric space.
**Theorem.** *The asymptotic dimension of the leaves of a foliated space is essentially constant.*

**Acknowledgments.** The work of the first author was supported by DGICYT Grant PB95-0850, and that of the second by NSF Grants No. DMS-9973086 and DMS-0049077. The authors were at the Universidade de Santiago de Compostela and at Caltech and CSUN while this paper was being written, and thank these institutions for their support. We also thank F. Alcalde Cuesta for helpful conversations.
# Chapter 2

# Foliated Spaces and Pseudogroups

This chapter collects and develops some basic results on foliated spaces and pseudogroups of local homeomorphisms that will be used in the rest of the monograph. General references on foliated spaces are [?], [?], [4].

## 2.1 Topological Preliminaries

This section collects several definitions and theorems on topological spaces that will be used subsequently in this monograph and particularly in this chapter.

**Theorem 2.1.** A space is metrizable if and only if it admits an embedding into a (generalized) Hilbert space.

A separable and metrizable space is homeomorphic to subset of Hilbert space.

**Definition 2.2.** A topological space is paracompact if it is Hausdorff and every open cover has a locally finite open refinement.

**Proposition 2.3.** (i) Every paracompact space is normal.

(ii) Every regular and second countable space is paracompact.

(iii) A second countable paracompact space is metrizable.

A partition of unity on a topological space $X$ is a collection of functions $\phi : X \to [0, 1]$ such that

* it is locally finite;

* for each $x$ in $X$, $\sum \phi(x) = 1$

If $\mathcal{U}$ is an open cover of $X$ and $\Phi$ is a partition of unity, then $\Phi$ is said to be subordinate to the cover $\mathcal{U}$ is for each $\phi \in \Phi$ there is an open set $U \in \mathcal{U}$ such that $\phi \equiv 0$ on $X \setminus U$.

**Theorem 2.4.** Every metric space is paracompact.
**Proposition 2.5.** Every open cover of a metric space has a partition of unity subordinate to it.

A function $f: X \to [-\infty, \infty]$ on a topological space is lower semicontinuous if $f^{-1}(a, \infty)$ is open in $X$ for each $a \in \mathbb{R}$.

**Definition 2.6.** A topological space is called a Polish space if its topology is second countable and induced by a complete metric.

**Proposition 2.7.** A subset of a polish space is a polish space if and only if it is a $G_\delta$-set.

## 2.2 Foliated spaces

The definition of the concept of foliated space requires that of smooth function in an appropriate setting. Let $Z$ be a topological space and let $U$ be an open set in $\mathbb{R}^n \times Z$. A map $f: U \to \mathbb{R}^p$ is smooth (of class $C^r$) at the point $(x_0, z_0)$ in $U$ if there is a neighborhood of $(x_0, z_0)$ in $U$ of the form $D \times Y$ (where $D$ is an open ball in $\mathbb{R}^n$ and $Y$ is an open subset of $Z$) such that $f$ is continuous on $D \times Y$ and the partial derivatives up to order $\le r$ of all coordinate functions $f_i$ of $f$ exist and are continuous at all points $(x, z) \in D \times Y$. The map $f$ is smooth on $U$ if it is smooth at each point of $U$. A map $f: U \to \mathbb{E}$ from $U$ into the Banach space $\mathbb{E}$ is smooth of class $C^r$ if each point of $U$ has a neighborhood of the form $D \times Y$, where $D$ is an open ball in euclidean space and $Y$ is a subset of $Z$, such that the assignement $y \in Y \mapsto f(\cdot, y) : D \to \mathbb{E}$ induces a continous map $Y \to C^r(D, \mathbb{E})$, where $C^r(D, \mathbb{E})$ is the space of maps of class $C^r$ of $D$ into $\mathbb{E}$, endowed with the topology of uniform convergence on compact sets. A foliated structure $\mathcal{F}$ of dimension $n$ and class $C^r$, $0 \le r \le \infty$, on a topological space $X$ is given by a maximal collection of pairs $(U_i, \phi_i)$, where $\{U_i\}$ is a covering of $X$ by open sets, and the maps $\phi_i$ are homeomorphisms $\phi_i: U_i \to D_i \times Z_i$, where $D_i$ is an open ball in euclidean space $\mathbb{R}^n$ and $Z_i$ is a topological space, and such that the coordinate changes corresponding to an intersection $U_i \cap U_j$ are of the form

$$ \phi_i \circ \phi_j^{-1}(x_j, z_j) = (x_i(x_j, z_j), z_i(z_j)), $$

where $x_i : \phi_j(U_i \cap U_j) \to \mathbb{R}^n$ is of class $C^r$ and $z_i$ is continuous. Then $(X, \mathcal{F})$, or just $X$, is called a foliated space. The pairs $(U_i, \phi_i)$ are called foliation charts (or flow boxes); the sets $U_i$ are the domains of the foliation chart $\phi_i$, and are called distinguished open sets; and their subsets $\phi_i^{-1}(D_i \times \{z\})$ ($z \in Z_i$) are called plaques. A family $\{(U_i, \phi_i)\}$ for a foliated space that satisfies all the conditions listed above, expect perhaps the maximality condition, is called a foliated atlas of the foliated space.

**Remark.** Note that the usual definitions of foliated spaces include the condition that the underlying topological structure of the foliated space be locally compact and metrizable. Local compactness is not assumed here because there is an important family of foliated spaces that are not locally compact. Metrizability cannot be ignored for too long.
The foliated structure of a foliated space $X$ induces a locally euclidean topology on $X$ for which the basic open sets are the open subsets of the plaques of a foliated atlas for $X$. This locally euclidean topology is finer than the original topology; to distinguish (verbally) between these topologies on $X$, they are sometimes called *ambient space topology* (the original one) and *leaf topology* (the locally euclidean one). The connected components of $X$ in the leaf topology are called *leaves*; the plaques in a flow box $(U, \phi)$ are the connected components of the intersection of $U$ with the leaves. The smooth structure implies that each leaf is a connected manifold of dimension $n$ and of class $C^r$. If $x$ is a point of $X$, the leaf which contains $x$ will usually be denoted by $L_x$. Concepts of manifold theory readily extend to foliated spaces. In particular, if $\mathcal{F}$ is at least of class $C^1$, there is a continuous vector bundle $TX$ over $X$ whose fiber at each point $x$ of $X$ is the tangent space of the manifold $L_x$ at $x$. Some basic properties of the structure of foliated spaces will now be studied. Most of these properties have already been established in other places under the additional assumption that the foliated space is locally compact. Let $X$ be a foliated space. A foliated chart $(U, \phi_U)$, with coordinate function $\phi_U: U \to D_U \times T_U$, is said to extend the foliated chart $(V, \phi_V)$ if $\overline{V} \subset U$ and if there is a disk $D_V \subset D_U$ and a subspace (open, a fortiori) $T_V \subset T_U$ such that the coordinate map of $V$ is $\phi_V: V \to D_V \times T_V$ and given by $\phi_V = \phi_U|_V$. If $X$ is a foliated space whose underlying topological structure is paracompact, then given an open cover $\mathcal{U} = \{U_i, \phi_i\}$ of $X$ by foliated charts, it is always possible to find a locally finite cover $\{V_\alpha, \phi_\alpha\}$ which is a *refinement* of $\mathcal{U}$ in the sense that each $\overline{V_\alpha}$ is contained in some $U_{i(\alpha)}$ and the corresponding chart $\phi_{i(\alpha)}$ extends $\phi_\alpha$. A foliated chart $(V, \phi)$ is said to be *uniform* in another foliated chart $(U, \phi)$ when $V \subset U$ and every plaque in $U$ meets at most one plaque in $V$. An atlas $\{U_i, \phi_i\}$ of $X$ is called *regular* if, whenever $U_i$ meets $U_j$, there is some foliated chart for $X$ where both $(U_i, \phi_i)$ and $(U_j, \phi_j)$ are uniform; in particular, each plaque in $U_i$ meets at most one plaque in $U_j$. The usual arguments for foliated manifolds show that there exist arbitrarily fine locally finite regular atlases of foliated spaces [?],[?], [?].

**Definition 2.8.** Let $X$ be a foliated space. A foliated atlas $\mathcal{U} = \{(U_\alpha, \phi_\alpha)\}$ is said to be regular if

(i) For each $\alpha$ there is a foliation chart $(W_\alpha, \psi_\alpha)$ for $X$ such that $\overline{U_\alpha} \subset W_\alpha$.

(ii) The cover $\{U_\alpha\}$ is locally finite.

(iii) If $U_\alpha$ and $U_\beta$ are domains of charts from $\mathcal{U}$, then the interior of each closed plaque $P \subset \overline{U_\alpha}$ meets at most one plaque in $\overline{U_\beta}$.

**Proposition 2.9.** Let $X$ be a locally compact foliated space. If $\mathcal{W}$ is covering of $X$ by domains of foliation charts, then there is a regular foliation atlas whose domains of foliation charts are relatively compact and form a covering of $X$ subordinate to the covering $\mathcal{W}$.

*Proof.* Cf. [?, Lemma 1.2.17] □

<figure><img src="image_2.png" /></figure>

**Proposition 2.10.** Let $X$ be a foliated space whose underlying topological structure is separable and metrizable. If $\mathcal{W}$ is an open covering of $X$ by domains of foliation charts, there is a regular foliation atlas whose domains form an open covering of $X$ subordinate to $\mathcal{W}$.
*Proof.* Because a metrizable space is paracompact, the atlas $\mathcal{W} = \{(W_\alpha, \phi_\alpha)\}$ is assumed to be such that the domains $W_\alpha$ of its foliation charts form a locally finite covering of $X$.

For each $x \in X$ choose an index $\alpha(x)$ such that $x \in W_{\alpha(x)}$ and choose a foliation chart $(U_x, \varphi_x)$ which is subordinate to the chart $(W_{\alpha(x)}, \phi_{\alpha(x)})$, such that $x \in U_x$ and that $\overline{U_x}$ is contained in the finitely many domains $W_\alpha$ which contain $x$.

Suppose that $U_x \subset W_\beta$, $\beta \neq \alpha(x)$, and write $\phi_\beta = (x_\beta, z_\beta)$ for the coordinate map, where

$$z_\beta : W_\beta \to Z_\beta$$

is the transverse coordinate map. By choosing $U_x$ to be a smaller neighborhood of $x$ (and maintaining at the same time the property that $(U_x, \phi_x)$ is subordinate to $(W_{\alpha(x)}, \phi_\alpha)$) it may be assumed that the fibers of $z_\beta : U_x \to Z_\beta$ are connected.

Because each $U_x$ is contained in at most finitely many of the domains $W_\beta$, this process is repeated finitely many times to ensure that whenever $U_x \subset W_\beta$, each plaque of $W_\beta$ contains at most one plaque of $U_x$.

The collection of domains $\{U_x\}$ forms a covering of $X$ by foliated charts, which satisfies (i) of Definition 2.8 by constrution.

It also satisfies property (iii). Indeed, if $U_x \cap U_y \neq \emptyset$, then $U_x \cap U_y$ is contained in finitely many domains $W_\alpha$. If $W_\alpha$ is any of those, then distinct plaques of $\overline{U_x}$ (respectively $U_y$) lie in distinct plaques of $W_\alpha$. Hence, each plaque of $U_x$ has interior meeting at most one plaque of $U_y$, and vice versa.

Because $X$ is separable and metrizable, $\{U_x\}$ has a locally finite refinement $\square$

**Lemma 2.11.** Let $X$ be a foliated space whose underlying topological structure is normal. Let $(U, \phi)$ be a foliation chart in $X$ and let $V \subset W \subset U$ be domains of subordinated foliation charts. Then there is a smooth function $f : X \to \mathbb{R}$ such that $f > 0$ on $V$ and $f \equiv 0$ outside $U$.

*Proof.* Because the foliated charts are subordinated, $\overline{V} \subset W$ and $\overline{W} \subset U$. Let $\phi : U \to D \times Z$ be the coordinate map of the chart $(U, \phi)$. Then $\phi(V)$ is of the form $D_V \times Z_V$, where $D_V$ is an open disk with compact closure contained in $D_U$, because $V$ is subordinate to $U$, and similarly $\phi(W) = D_W \times Z_W$, where $D_W \subset D_U$ is an open disk such that $\overline{D_V} \subset D_W$. Since $\overline{V} \subset U$ and $\phi$ is a homeomorphism, the image $\phi(\overline{V}) = \overline{D_V} \times \overline{Z_V}$. Similarly, $\phi(W) = D_W \times Z_W$ contains the closure of $\phi(V)$, and the relation between the various subsets is that $D_V \subset D_W \subset D_U$, with each disk $D_V$ and $D_W$ having compact closure in the next larger disk $D_W$ and $D_U$, respectively, and transversally $Z_V \subset \overline{Z_V} \subset Z_W \subset \overline{Z_W} \subset Z_U$

Since $Z_U$ is a normal topological space, there is a continuous function $f_Z : Z_U \to \mathbb{R}$ such that $f_Z \equiv 1$ on $Z_V$ and $f_Z \equiv 0$ outside $Z_W$. There is also a smooth function $f_D$ on $D_U$ such that $f_D \equiv 1$ on $D_V$ and $f_D \equiv 0$ outside $\overline{D_W}$.

Then $f = (f_D \times f_Z) \circ \phi$ is a smooth function of class $C^r$ on $U$ which is $> 0$ on $V$ and identically equal to $0$ outside $W$. Therefore $f$ extended by $0$ outside $U$ is a smooth function on $X$ with the properties desired. $\square$
**Lemma 2.12.** Let $X$ be a foliated space whose underlying topological structure is paracompact. Let $U \subset X$ be the domain of a foliation chart and let $C$ be a closed subset of $X$ such that $C \subset U$. Then there is a smooth function $f$ on $X$ such that $f > 0$ on $C$ and $f \equiv 0$ outside $U$.

*Proof.* Note that a paracompact space is normal, and a normal space admits a base of closed neighborhoods for its topology.

Let $\phi : U \to D \times Z$ be the coordinate function of $U$, where $D$ is a disk in euclidean space. Let $p$ be a point in $U$. Because the topology of $\phi(U) = D \times Z$ has a basis consisting of open sets of the form $D' \times Z'$, where $D' \subset D$ is an open disk contained in $D$, and because the topology of $X$ is normal, there is an open set $D' \times Z'$ in $\phi(U)$ which contains $\phi(p)$ and such that the closure of $\phi^{-1}(D' \times Z')$ in $X$ is contained in the interior of a closed neighborhood of $x \in X$ which in turn is contained in $U$.

Since $C$ is closed in $X$, it inherits the paracompact property of $X$. Thus there is a locally finite family of foliation charts $\mathcal{V} = \{V_i\}$ which form an open covering of $C$. By Lemma 2.11, for each index $i$ there is a smooth function of class $C^r$ $f_i$ on $X$ which is $> 0$ on $V_i$ and has support contained in $U$.

Since the covering $\mathcal{V}$ of $C$ is locally finite, the function $f = \sum_i f_i$ is well defined and of class $C^r$ on $X$ and has the desired property of being $> 0$ on $C$ and $\equiv 0$ outside of $U$. $\square$

The following theorem plays an key role in constructing global object on foliated spaces by gluing local ones. A proof in the locally compact case is given in Moore-Schochet ??.

**Theorem 2.13.** ?? Let $X$ be a foliated space of class $C^r$ whose underlying topology is paracompact, and let $\mathcal{U}$ be an open covering of $X$. Then there is a partition of unity $\{\phi_i\}$ subordinated to $\mathcal{U}$, and such that each function $\phi_i$ is of class $C^r$.

*Proof.* Because $X$ is paracompact, the original cover $\mathcal{U} = \{U_\alpha | \alpha \in A\}$ may be assumed to be locally finite, and because there is a base for the topology of $X$ consisting of foliated charts, the cover $\mathcal{U}$ can further be assumed to consist of foliated charts.

Consider families $F = \{g_\beta | \beta \in A_F\}$, where $A_F \subset A$ is a subset of indices, $g_\beta : X \to [0, 1]$ is a $C^r$-function such that if $W_\beta = \{x \in X | g_\alpha(x) > 0\}$, then $\overline{W_\beta} \subset U_\beta$ for every $\beta \in A_F$, and $\{W_\beta | \beta \in A_F\} \cup \{U_\alpha | \alpha \in A \setminus A_F\}$ is a covering of $X$.

The collection of these families is ordered by inclusion. Let $\mathbf{F}$ be a maximal chain and let $G = \bigcup F \in \mathbf{F}$. Then $\{W_\beta | \beta \in G\}$ is a covering of $X$ which refines $\mathcal{U}$ as desired. Thus every chain in $\mathbf{F}$ has an upper bound and by Zorn's lemma the collection of the families has a maximal element. Let this maximal element be of the form $\{W_\beta | W_\beta \subset U_\beta\}$ where $\beta \in B \subset A$. If $B \neq A$, then there is an index $\alpha \in A \setminus B$.

Let $\mathbf{F}$ be a maximal chain and $G = F \in \mathbf{F} \cup F$. Then $G = \{g_\beta | \beta \in A_G = \bigcup A_F$ for $F \in \mathbf{F}\}$. If $G \notin \mathbf{F}$, then there is a point $x \in X$ which is not in any $W_\beta$ for $\beta \in A_G$ or in $U_\alpha$ for any $\alpha \in A \setminus A_G$. Let $U_{\alpha_i}, i = 1, \dots, s$, be the finitely many elements of the cover $\mathcal{U}$ which contain $x$. Then each index $\alpha_i$ is in $A_G$, and since $\mathbf{F}$ is a chain, there is an index set $A_F$ which contains all the finitely many indices $\alpha_i$. Thus $x$ must be in some $W_{\alpha_i}$, a contradiction.
Next we show that $A_G = A$. If not, then there is an index $\gamma$ in $A$ but not in $A_G$. Let $Y \subset X$ be defined by

$$Y = \bigcup_{\beta \in A_G} W_\beta \cup \bigcup_{\alpha \in A \setminus A_G, \alpha \neq \gamma} U_\alpha$$

Then $Y$ is an open subset and $X \setminus Y$ is a closed subset of $X$ lying inside $U_\alpha$. Because $U_\alpha$ has the structure of a foliated chart with $\phi_U : U \to D \times T$, the image $\phi_U(X \setminus Y)$ is a closed subset of $D \times T$.

By Lemma 2.12, there is a $C^r$-function $g : X \to [0, 1]$ such that $g > 0$ on $C$ and $g \equiv 0$ outside an open set whose closure in $X$ is contained in $U_\alpha$. Therefore the family $G \cup \{g_\alpha\}$ properly contains $G$, contradicting the maximality of $G$.

It follows that the collection of functions $\{g_\alpha | \alpha \in A\}$ is such that $W_\alpha = \{g_\alpha(x) > 0\}$ is an open covering of $X$ with $\overline{W}_\alpha \subset U_\alpha$. The function $g = \sum_\alpha g_\alpha$ is well defined because of the local finiteness of the covering $\{W_\alpha\}$, and satisfies $g(x) > 0$ for all $x$ in $X$. Therefore $\{f_\alpha = g_\alpha/g\}$ is the desired partition of unity. $\square$

**Proposition 2.14 (**********).** Let $X$ be a foliated space of class $C^r$, $r \ge 1$, whose underlying topology is paracompact and separable. Then if $\mathcal{U}$ is a covering of $X$ by domains of foliation charts, then there is a subordinate

One application of the existence of smooth partition of unity on a foliated space is the following embedding theorem for compact foliated spaces.

**Theorem 2.15.** If $X$ is a compact foliated space of class $C^r$, then there is a smooth $C^r$ embedding $\varphi$ of $X$ in the separable real Hilbert space $\mathbb{E}$.

*Proof.* Let $\mathcal{U} = \{(U_i, \varphi_i)\}_{i=1}^N$ be a regular open cover of $X$ by foliation charts, where

$$\varphi_i : U_i \to D \times Z_i$$

and $D$ is a bounded open ball in $\mathbb{R}^n$. The metrizable spaces $Z_i$ embed in $\mathbb{E}$ because they are second countable [?, p. 195]. Hence the maps $\varphi_i : U_i \to D \times Z_i \subset \mathbb{R}^n \times \mathbb{E}$ can be assumed to be $C^r$-embeddings. Let $\{\lambda_i\}_{i=1}^N$ be a partition of unity subordinate to $\mathcal{U}$ and, for each $i$, define a $C^r$ mapping $h_i : M \to \mathbb{R}^n \times \mathbb{E}$ by

$$h_i(w) = \begin{cases} \lambda_i(w)\varphi_i(w) & \text{if } w \in U_i, \\ 0 & \text{otherwise.} \end{cases}$$

Let $\varphi : X \to (\mathbb{R}^n \times \mathbb{E})^n \times \mathbb{R}^n \cong \mathbb{E}$ be the map

$$\varphi(w) = (h_1(w), \dots, h_n(w), \lambda_1(w), \dots, \lambda_n(w)).$$

If $\varphi(w) = \varphi(u)$, then $\lambda_i(w) = \lambda_i(u) \neq 0$, for some index $i$; hence $w, u \in U_i$ and $\varphi_i(w) = \varphi_i(u)$. That is, $w = u$. Also, for each $w \in X$, there is an index $i$ such that $\lambda_i(w) \neq 0$; so $h_i$ restricts to an imbedding of class $C^r$ of a neighborhood of $w$ in the plaque through $w$ and $\varphi$ does also. Because $X$ is compact, the continuous injection $\varphi$ is a homeomorphism onto its image. $\square$
A relative version of the theorem will be used below and it is proved similarly.

**Corollary 2.16.** Let $X$ be foliated space whose underlying topology is separable and metrizable. If $K \subset X$ is a compact subset of $X$, then there is a smooth map $f: X \to E$ which is an embedding of a neighborhood of $K$ in $X$ into $E$.

**Proposition 2.17.** Let $X$ be a compact foliated space of class $C^r$, for some $r \ge 1$. Given a Riemannian metric $g$ on the leaves of $X$, there is a distance function on $X$ that induces the Riemannian distance on each leaf.

*Proof.* By Theorem ??, consider $X$ as embedded in $E$ as a compact subset. The tangent spaces to $E$ are given the natural inner product of the Hilbert space. The bundle $TX$ is a finite-dimensional subbundle of $T(E)|_X$, and it therefore has an orthogonal complement that we denote by $T^\perp X$. If $TX$ is given a positive definite inner product, then the direct sum decomposition $T(E)|_X = TX \oplus T^\perp X$ is given the inner product that is the direct sum of $g$ and the Hilbert inner product. This provides a section over $X$ of the bundle of positive definite symmetric bilinear forms on $T(E)|_X$. Since the fibers of this bundle are absolute neighborhood retracts and $X$ is compact, this section extends over all of $E$. This defines a distance function on $E$ that induces the given distance along the leaves. $\square$

A given metric tensor along the leaves of $X$ admits an extension to a metric tensor on $E$. This is so because a metric tensor may be viewed as a section of some bundle over $E$ with contractible fibers, and $X$ being closed in $E$ implies that a section over $X$ extends to one over $E$. It may only be continuous, of course, but that is sufficient to define length of paths on leaves of $X$. This embedding of a foliated space $X$ in $E$ also gives rise to a normal vector bundle over $X$ with fiber $E$. This structure permits to formalize concepts and results like “local projection of leaves onto leaves” or “Reeb’s stability.” This structure will be described presently. Let $X$ be a compact foliated space, of class $C^r$ with $r \ge 1$, embedded in the hilbert space $E$. The restriction of the embedding to each leaf is not an embedding, but only an injective immersion. Because $X$ is smooth of class $C^1$, the map that assigns to a point $x \in X$ the subspace $T_x X$ of $E$ is continuous (as a map of $X$ into the space of $n$-dimensional subspaces of $E$). It follows that if $F$ is a subspace complementary to one $T_x X$ in $E$, then it is also complementary to $T_y X$ for $y$ sufficiently close to $x$. The key point is that each tangent space $T_x X$ is a finite dimensional subspace of $E$, hence is closed and has an orthogonal complement. If $i: L \to E$ is the inclusion of a leaf of $X$ in $E$, then there are charts about $x$ in $L$ and $E$ such that the corresponding local representation of $i$ is of the form $y \mapsto (y, 0)$, provided by the implicit function theorem (flat coordinates). In these flat coordinates, $T_y X^\perp = \{y\} \times E$ for $y$ in this plaque containing $x$, and by continuity, the affine subspaces $y + T_y X^\perp$ meet nearby leaves transversely. Since $X$ is compact, it follows that this holds for all $y$ in a neighborhood of radius $r$ about $x$ in $X$, the radius $r$ being independent of the point $x$.

**Theorem 2.18.** Let $X$ be a compact foliated space embedded in Hilbert space $E$, of class $C^2$. Let $i: L \to E$ denote the inclusion of a leaf $L$ in $X \subset E$. Then there is a vector bundle $p: N \to L$ and a neighborhood $W$ of the zero section of $N$ such that the following hold:
(1) The map $i: L \to \mathbb{E}$ extends to a local diffeomorphism $\varphi: W \to \mathbb{E}$;

(2) there is a foliated space $Y \subset W$, of the same dimension as $X$, having $L$ as a leaf and transverse to the fibers of $N$; and

(3) as foliated spaces, $Y = \varphi^{-1}(X \cap \varphi(W))$ and the restriction of $p$ to each leaf of $Y$ is a local diffeomorphism into $L$.

Let $N(X) = \{(x,v) \mid v \perp T_x X\} \subset X \times \mathbb{E}$. Then the exponential map $\lambda : N(X) \to \mathbb{E}$ is $\lambda(x,v) = x + v$. Let $N(X, \varepsilon)$ denote the set of pairs $(x,v) \in N(X)$ with $\|v\| < \varepsilon$. The normal bundle $N(L)$ to each leaf $L$ is contained in $N(X)$. The differential $\lambda_*$ at each point $(x,0) \in N(X)$ is the identity (under the canonical identification $T_{(x,0)}N(X) = T_x X \times_{T_x X} X^\perp$). Therefore, by continuity of $\lambda_*$ on $N(X)$ and compactness of $X$, there exists $\varepsilon > 0$ such that $\lambda_*$ is an isomorphism at $(x,v)$ for $x \in X$ and $\|v\| < \varepsilon$. This property of $\lambda_*$ implies the following lemma.

**Lemma 2.19.** For each point $x \in X$ there exists a neighborhood $U_x$ in the leaf through $x$ such that $\lambda$ is a diffeomorphism of $N(U_x, \varepsilon)$ into $\mathbb{E}$.

A corollary is that the compactness of $X$ implies that the statement of the lemma holds uniformly on $X$.

**Corollary 2.20.** There exists $\varepsilon > 0$ and $r > 0$ such that $\lambda$ is a diffeomorphism of $N(B(x, r), \varepsilon)$ into $\mathbb{E}$, for every $x \in X$.

*Proof.* Otherwise, by compactness of $X$, there exist sequences $(x_n, v_n) \neq (y_n, w_n)$ in $N(X)$, with $\|v_n\|, \|w_n\| \to 0$, $x_n, y_n \to x$, and such that $\lambda(x_n, v_n) \neq \lambda(y_n, w_n)$ for each $n$. Working on a local flow box around $x$ in $X$, choosing flat coordinates around the plaque through $x$, and taking into account that the normal subspace to points of $X$ varies continuously, it is then obvious that it is possible to find new sequences $(x'_n, v'_n)$ and $(y'_n, w'_n)$ as above with the added property that all the points $x'_n$ and $y'_n$ are in the same plaque as $x$. This contradicts Lemma 2.19. $\square$

The vector bundle $N(X)$ has a metric on the fibers so that the continuous map $\phi(x,v) = \lambda(x,v) - x$ is a linear isometry on each fiber. By compactness of $X$, there exists $\varepsilon_0 > 0$ such that for each leaf $L$ of $X$ the restricted map $\lambda : N(L, \varepsilon_0) \to \mathbb{E}$ is a local diffeomorphism. Moreover, there exists $r > 0$ such that for every leaf $L$ and every point $x \in L$, the map $\lambda : N(B(x,r), \varepsilon_0) \to \mathbb{E}$ is a diffeomorphism. It follows that if $L$ is a leaf of $X$, then the neighborhood $N(L, \varepsilon_0)$ contains a foliated space $Y$, which is lifted from $X$ via the local diffeomorphism $\lambda$. This completes the proof of Theorem 2.18.

**Remark.** $\mathbf{******C}^3$?? The hypothesis that $X$ be of class $\mathbf{C}^2$ is required so that the normal bundle and exponential map be of class $\mathbf{C}^1$. On the other hand, a manifold of class $\mathbf{C}^1$ admits a compatible structure of class $\mathbf{C}^r$, for any $1 \le r \le \infty$. The proof (see [?, Chapter 2]) is based on approximation results that can be adapted to ‘manifolds with parameters’, that is, to foliated spaces. Since it is out of place to do this here, the $\mathbf{C}^2$ hypothesis is kept here and in Theorems ??, ??
**Remark.** Let $x \in L$, and let $T$ be a transversal through $x$, which may be taken to lie in the affine subspace $x + T_x X^{\perp}$ of $\mathbb{E}$. If $\mathcal{G}(x)$ denotes the group of germs of local homeomorphisms of $T$ which fix $x$, then there is the germinal holonomy representation

$$ \pi_1(L, x) \to \mathcal{G}(x). $$

The construction above shows that this representation is equivalent to that of $L$ as a leaf of the foliated space $Y$ of $N(L, \varepsilon_0)$.

The following two propositions complement Theorem 2.18.

**Proposition 2.21.** Let $L$ be a leaf of the compact foliated space of Theorem 2.18. Then there exists $\varepsilon > 0$ and an $\varepsilon$-disc bundle $W \to L$ which carries a foliated space $Y$ and there exists a constant $K > 0$ such that the projection $p: S \to L$ of each leaf $S$ of $Y$ into $L$ is a local diffeomorphism (of class $C^1$) of distortion in the interval $[1/K, K]$.

The disk bundle $W \to L$ and foliated space $Y \subset W$ with the properties stated above have been constructed in the proof of the previous proposition. Because $X$ is compact, it admits a finite regular cover by foliation charts. The projection $Y \to L$ amounts to finitely many projections between plaques of these foliation charts. Because the cover is finite and regular, there are global bounds for the distortion of these projections within each fiation chart. The claim made in the proposition follows from these observations.

**Proposition 2.22.** Let $L$ be as in Proposition 2.21. Given $R > 0$, there exists $\delta > 0$ such that, if $x \in L'$ and $y$ is a point of $W_0$ with $p(y) = x$, and such that the distance in the fiber of $W_0$ between $x$ and $y$ is $< \delta$, then the ball $B(y, R)$ in the leaf of $Y$ through $y$ is contained in $W_0$.

*Proof.* The proof of this fact uses again a finite regular cover by foliation charts. The local coordinate changes in the transverse direction are uniformly continuous maps, and only a finite number of coordinate changes are required to run through a ball of radius $R$, because plaques have a definite size. $\square$

**Remark.** The construction of Theorem 2.18 and the discussion that follows it (especially Proposition 2.21), extend to a covering $\pi: L' \to L$ of a leaf $L$, simply by considering the normal bundle of the immersion $L' \to L \hookrightarrow \mathbb{E}$. The output is a disc bundle $W \to L'$ containing a foliated space $Y$ transverse to the fibers and having $L'$ as a leaf. The projection of leaves of $Y$ into $L$ is a local diffeomorphism of bounded distortion.

Keeping the notation introduced so far, let $L' \to L$ be a covering of a leaf $L$ of $X$, let $x \in L'$ and let $D$ be a compact manifold with boundary. Let $y \in D$ and let $f: D \to L'$ be a continuous map with $f(y) = x$. The following is a Reeb stability type of result.

**Proposition 2.23.** Suppose that the induced homomorphism $\pi_1(D, y) \to \pi_1(L', x)$ takes $\pi_1(D, x)$ into the kernel of the holonomy representation of $L'$ (as a leaf of the foliated space $Y$). Then there is a transversal $Z \subset Y$ through $x$ and a smooth map $F$ from the product foliation $D \times Z$ into $X$ such that $F|_{D \times x} = f$ and $F(D \times y)$ is contained in the leaf through $y$, for all $y \in Z$.
If $f: D \hookrightarrow L$ is an embedding, $Z$ can be chosen so that $F$ embeds $D \times Z$ in $Y$. In particular, if $\pi_1(L', x) \to \pi_1(L)$ has image contained in the kernel of the holonomy representation of $L$ as a leaf of $X$, then, given $x \in L'$ and $R > 0$ arbitrarily large, there exists $\delta > 0$ such that if $y$ is a point in the fiber $p^{-1}(x)$ at distance $< \delta$ from $x$, then the ball $B(y, KR)$ contains $p^{-1}(B(x, R)) \cap Y$, and the component of this last set which contains $y$, contains the ball $B(y, R/K)$. Moreover, because there is no holonomy, it is possible to choose $\delta$ so that there is a transversal $Z$ in the fiber through $x$ such that the union of the leaves of $p^{-1}(B(x, R)) \cap Y$ through points of $Z$ is parametrized as a product $B(x, R) \times Z$, and the induced metric on the leaves is at bounded distance from that of $B(x, R)$. Another consequence of the existence of partitions of unity is that a foliated space of class $C^r$, with $r \ge 1$, and whose underlying topological structure is paracompact and normal, always admits a metric tensor; i.e., an inner product on $TX$ inducing a riemannian metric on each leaf, and varying continuously from leaf to leaf in the appropriate way. The induced distance on a leaf $L$ of $X$ will be denoted by $d_L$. If $X$ is compact, then each leaf is a complete riemannian manifold (and of bounded geometry, a fortiori). The completeness property of the leaves does not necessarily hold in a general foliated space, but it can always be arranged to be so after a conformal change of metric.

**Theorem 2.24.** Let $X$ be a foliated space of class $C^r$, $r \ge 1$, whose underlying topology is metrizable and separable. Then there is a metric tensor on $X$ for which every leaf is a complete Riemannian manifold.

*Proof.* The corresponding theorem for manifolds is due to Nomizu-Ozeki [?]; only the details which are proper to foliations will be discussed in this proof.

As remarked above, there is a metric tensor $g$ on $X$. With respect to the metric induced on the leaves of $X$ by such metric tensor, define $r(x)$ to be the supremum of all real numbers $r > 0$ such that the metric ball $B(x, r)$ in the leaf through $x$ is relatively compact in that leaf.

**Claim 1.** The function $r: X \to (0, \infty]$ is lower semicontinuous on $X$, that is, if $x_n \to x$ in $X$, then $\liminf r(x_n) \ge r(x)$.

If that was not the case, then there would exist a sequence $x_n$ converging to $x$ and such that $\liminf r(x_n) < r(x)$. Thus $\liminf r(x_n) < \infty$ and there would be an $\varepsilon > 0$ such that $r(x_n) + 2\varepsilon < r(x)$ for all $n$. Because the exponential mapping $\exp_x: T_xX \to X$ is defined on the closed ball of radius $r(x) - \varepsilon$ and center $0_x$ in the tangent space $T_xL_x$. This would imply [****] that for $n$ sufficiently large, $r(x_n) \ge r(x) - \varepsilon$, a contradiction. It follows that $r$ is lower semicontinuous.

Let $U_s = \{x \in X \mid r(x) > s\}$. Then $\{U_s \mid s > 0\}$ is an open covering of $X$ because $r$ is lower semicontinuous and $r(x) > 0$ for all $x \in X$. If $\{\phi_s\}$ is a partition of unity (smooth of class $C^r$) subordinated to the covering $U_s$ (which exists because $X$ is paracompact and thus Theorem ?? applies), then $h = \sum_{s>0} s\phi_s$ is a function on $X$ of class $C^r$ such that $0 < h < r$.

Then the metric tensor $(1/h)^2 \cdot g$ induces a complete Riemannian metric on each leaf of $X$. This is proved exactly as in Nomizu-Ozeki [?], by showing that, for any $x$ in $X$, the ball of
radius 1/3 and center $x$ in the leaf $L_x$ in the new metric is contained in the ball $B(x, r(x)/2)$ (for the old metric), which is compact. □

## 2.3 Pseudogroups of local transformations

A pseudogroup of local transformations of a topological space $Z$ is a collection $\mathcal{H}$ of homeomorphisms between open subsets of $Z$ that contains the identity on $Z$ and is closed under composition (wherever defined), inversion, restriction and combination of maps. Such a pseudogroup $\mathcal{H}$ is generated by a set $E \subset \mathcal{H}$ if every element of $\mathcal{H}$ can be obtained from $E$ by using the above pseudogroup operations; to simplify arguments, the sets of generators to be considered will be assumed to be symmetric ($h^{-1} \in E$ if $h \in E$). The orbit of an element $x \in Z$ is the set $\mathcal{H}(x)$ of elements $h(x)$, for all $h \in \mathcal{H}$ whose domain contains $x$. These orbits are the equivalence classes of an equivalence relation on $Z$. Note that an arbitrary equivalence relation $R \subset Z \times Z$ is defined by a pseudogroup on $Z$ if and only if $R$ is a union of sets $R_i$ such that the restriction to each $R_i$ of both factor projections $Z \times Z \to Z$ are homeomorphisms. Indeed, take the sets $R_i$ to be the graphs of all local transformations in the pseudogroup. Moreover $R$ is defined by a countably generated pseudogroup on $Z$ if and only if $R$ is a countable union of sets $R_i$ satisfying the above condition. This follows because a countable set of local transformations of $Z$ gives rise to a countable family of composites with maximal domain. The set of germs of all transformations in the pseudogroup $\mathcal{H}$ at all points of their domains, endowed with the étale topology, is a topological groupoid, product and inversion being induced by composition and inversion of maps, respectively. Thus, for each $x \in Z$, the set of germs at $x$ of transformations $h \in \mathcal{H}$ with $x \in \text{dom} h$ and $h(x) = x$ is a group called the group of germs at $x$. If $x, y \in Z$ are in the same $\mathcal{H}$-orbit, then the groups of germs at $x$ and $y$ are isomorphic: an isomorphism is given by conjugation with the germ at $x$ of any transformation $g \in \mathcal{H}$ with $x \in \text{dom} g$ and $g(x) = y$. The group of germs of an orbit is therefore well defined, up to isomorphisms, as the group of germs at any point of that orbit. In particular, a distinguished type of orbits are those with trivial group of germs. Pseudogroups of local transformations must be thought of as natural generalizations of group actions on topological spaces (the transformations of each group action generates a pseudogroup). But the main example to keep in mind is the representative of the holonomy pseudogroup of a foliated space $(X, \mathcal{F})$ associated to a regular atlas $\mathcal{U} = \{U_i, \phi_i\}$, whose construction is now recalled. Assume that the underlying topology of $X$ is metrizable and separable. Then there exist a foliation atlas $\{(U_i, \phi_i)\}$ for $X$ which is regular and countable. If $\phi_i: U_i \to D_i \times Z_i$ for metrizable and separable spaces $Z_i$ and open balls $D_i$ of finite radius in $\mathbb{R}^n$, let $p_i: U_i \to Z_i$ denote the composite of $\phi_i$ with the factor projection $D_i \times Z_i \to Z_i$; thus the fibers of each $p_i$ are the plaques in $U_i$. If $U_i$ meets $U_j$, let $Z_{i,j} = p_i(U_i \cap U_j)$, and the regularity of $\mathcal{U}$ implies the existence of a homeomorphism $h_{i,j}: Z_{i,j} \to Z_{j,i}$ such that $p_j = h_{i,j} \circ p_i$ on $U_i \cap U_j$ [4, ?, ?]. Then the collection $\{U_i, p_i, h_{i,j}\}$ is called a *defining cocycle* for $\mathcal{F}$ [15, ?]. These $h_{i,j}$ generate a pseudogroup $\mathcal{H}$ of local transformations of $Z = \bigsqcup_i Z_i$, which is called the representative of the holonomy pseudogroup of $(X, \mathcal{F})$ induced by $\mathcal{U}$ (or by $\{U_i, p_i, h_{i,j}\}$). Because the foliated atlas is countable and the topology of $X$ is separable
and metrizable, it follows that the holonomy pseudogroup is countable generated. There is a canonical bijection between the set of leaves and the set of $\mathcal{H}$-orbits, which is given by $L \mapsto \mathcal{H}(p_i(x))$ if $x \in L \cap U_i$. Each $Z_i$ can be considered as a local transversal of $\mathcal{F}$ via $\phi_i$ and the identification $Z_i \equiv \{y_i\} \times Z_i \subset D_i \times Z_i$ for any fixed point $y_i \in D_i$ (for instance, its center). It may be assumed that all of these local transversals are disjoint from each other, and thus that $Z$ is embedded into $X$ as a complete transversal of $\mathcal{F}$. Each $\mathcal{H}$-orbit injects into the corresponding leaf in this way. It is well known that all defining cocycles of a foliated space induce holonomy pseudogroups that are equivalent in the sense given by the following definition; thus the relevant properties of pseudogroups of local transformations of a topological space are those that are invariant by these equivalences.

**Definition 2.25** (Haefliger [15, ?]). Let $\mathcal{H}, \mathcal{H}'$ be pseudogroups of local transformations of topological spaces $Z, Z'$, respectively. An *étale morphism* $\Phi : \mathcal{H} \to \mathcal{H}'$ is a maximal collection $\Phi$ of homeomorphisms of open subsets of $Z$ to open subsets of $Z'$ such that:

* If $\phi \in \Phi$, $h \in \mathcal{H}$ and $h' \in \mathcal{H}'$, then $h' \circ \phi \circ h \in \Phi$;

* the sources of elements of $\Phi$ form a covering of $Z$; and

* if $\phi, \psi \in \Phi$, then $\psi \circ \phi^{-1} \in \mathcal{H}'$.

An *étale morphism* $\Phi : \mathcal{H} \to \mathcal{H}'$ is called an *equivalence* (or *étale equivalence*) if the collection $\Phi^{-1} = \{\phi^{-1} | \phi \in \Phi\}$ is also an *étale morphism* $\mathcal{H}' \to \mathcal{H}$, which is called the *inverse* of $\Phi$. The *composite* of two *étale morphisms* $\Phi : \mathcal{H} \to \mathcal{H}'$ and $\Psi : \mathcal{H}' \to \mathcal{H}''$ is the collection $\Psi \circ \Phi = \{\psi \circ \phi | \phi \in \Phi, \psi \in \Psi\}$, which is an *étale morphism* $\mathcal{H} \to \mathcal{H}''$. Finally, an *étale morphism* $\Phi : \mathcal{H} \to \mathcal{H}'$ is *generated* by a subset $\Phi_0 \subset \Phi$ if all the elements of $\Phi$ can be obtained by restriction and combination of composites $h' \circ \phi \circ h$ with $h \in \mathcal{H}, \phi \in \Phi_0$ and $h' \in \mathcal{H}'$.

An *étale morphism* $\Phi : \mathcal{H} \to \mathcal{H}'$ clearly induces a continuous map between the corresponding spaces of orbits, $\bar{\Phi} : Z/\mathcal{H} \to Z'/\mathcal{H}'$, which is a homeomorphism if $\Phi$ is an equivalence. A basic example of a pseudogroup equivalence is the following. Let $\mathcal{H}$ be a pseudogroup of local transformations of a space $Z$, let $U \subset Z$ be an open subset that meets every $\mathcal{H}$-orbit, and let $\mathcal{G}$ denote the restriction of $\mathcal{H}$ to $U$. Then the inclusion map $U \hookrightarrow Z$ generates an equivalence $\mathcal{G} \to \mathcal{H}$. In fact, this example can be used to describe any pseudogroup equivalence in the following way. Let $\mathcal{H}, \mathcal{H}'$ be pseudogroups of local transformations of spaces $Z, Z'$, and $\Phi : \mathcal{H} \to \mathcal{H}'$ an equivalence. Then let $\mathcal{H}''$ be the pseudogroup of local transformations of $Z'' = Z \cup Z'$ generated by $\mathcal{H} \cup \mathcal{H}' \cup \Phi$. Then the inclusions of $Z, Z'$ in $Z''$ generate equivalences $\Psi_1 : \mathcal{H} \to \mathcal{H}''$ and $\Psi_2 : \mathcal{H}' \to \mathcal{H}''$ so that $\Phi = \Psi_2^{-1} \circ \Psi_1$. For pseudogroups of local transformations of locally compact spaces, the following result characterizes the existence of relatively compact open subsets that meet all orbits.

**Lemma 2.26.** Let $\mathcal{H}$ be a pseudogroup of local transformations of a locally compact $Z$. (The orbit space $Z/\mathcal{H}$ is compact if and only if there exists a relatively compact open subset that meets every $\mathcal{H}$-orbit.
*Proof.* If an open subset $U \subset Z$ meets every $\mathcal{H}$-orbit, then the restriction $U \to Z/\mathcal{H}$ of the quotient map $Z \to Z/\mathcal{H}$ is onto. Thus $Z/\mathcal{H}$ is compact because it is a continuous image of the compact space $\bar{U}$.

Assume that $Z/\mathcal{H}$ is compact. Since $Z$ is locally compact, each $x \in Z$ has relatively compact open neighborhood $U_x$. Let $Q_x$ denote the image of $U_x$ by the quotient map $Z \to Z/\mathcal{H}$. Since $Z/\mathcal{H}$ is compact, its open covering $\{Q_x \mid x \in Z\}$ has a finite sub-covering $Q_{x_1}, \dots, Q_{x_m}$. Then the open set $U = U_{x_1} \cup \dots \cup U_{x_n}$ of $Z$ is relatively compact and meets all orbits of $\mathcal{H}$. $\square$

Examples of pseudogroups with compact space of orbits are the holonomy pseudogroups of compact foliated spaces, as orbit and leaf spaces can be identified. But such pseudogroups satisfy a stronger compactness condition that is defined as follows.

**Definition 2.27 (Haefliger [15]).** Let $\mathcal{H}$ be a pseudogroup of local transformations of a locally compact space $Z$. Then $\mathcal{H}$ is *compactly generated* if there is a relatively compact open set $U$ in $Z$ meeting each orbit of $\mathcal{H}$, and such that the restriction $\mathcal{G}$ of $\mathcal{H}$ to $U$ is generated by a finite symmetric collection $E \subset \mathcal{G}$ so that each $g \in E$ is the restriction of an element $\bar{g}$ of $\mathcal{H}$ defined on some neighborhood of the closure of the source of $g$.

Haefliger [15] observed that this concept of compactly generated pseudogroup is invariant by equivalences of pseudogroups, and that the relatively compact open set $U$ meeting each orbit can be chosen arbitrarily. If $E$ satisfies the conditions of Definition 2.27, it will be called a *system of compact generation* of $\mathcal{H}$ on $U$.
# Chapter 3

## Metric structures on leaves

### 3.1 Coarse quasi-isometries

Let $M, M'$ be arbitrary metric spaces with metrics $d, d'$. Recall that a map between metric spaces, $f: M \to M'$, is said to be *lipschitz* if there is some $C > 0$ such that

$$d'(f(x), f(x')) \le C d(x, y)$$

for all $x, y \in M$. Such a $C$ will be called a *lipschitz distortion* of $f$. The map $f$ is said to be *bi-lipschitz* when there is some $C \ge 1$ such that

$$\frac{1}{C} d(x, y) \le d'(f(x), f(x')) \le C d(x, y)$$

for all $x, y \in M$. In this case, the constant $C$ will be called a *bi-lipschitz distortion* of $f$. Any bi-lipschitz map is injective. A collection of lipschitz maps is called *equilipschitz* when they have some common lipschitz distortion. A collection of bi-lipschitz maps is said to be *equi-bi-lipschitz* when they have some common bi-lipschitz distortion, which is called an *equi-bi-lipschitz distortion*.

A *net* in a metric space $M$, with metric $d$, is a subset $A \subset M$ such that $d(x, A) \le K$ for some $K > 0$ and all $x \in M$; the term $K$-net is also used. On the other hand, a subset $A$ of $M$ is said to be *separated* when there is some $\delta > 0$ such that $d(x, y) > \delta$ for every pair of different points $x, y \in A$

**Lemma 3.1.** Let $K > 0$. There is some $K$-separated $K$-net of $M$. Moreover any $K$-net of $M$ contains a $K$-separated $2K$-net of $M$.

*Proof.* Let $\mathcal{S}$ be the family of $K$-separated subsets of $M$. By using Zorn lemma, it follows that there exists a maximal element $A \in \mathcal{S}$. If $d(x, A) > K$ for some $x \in M$, then $A \cup \{x\} \in \mathcal{S}$, contradicting the maximality of $A$. Hence $A$ is a $K$-net in $M$.

Now, let $A$ be a $K$-net of $M$. Then there is a $K$-separating $K$-net $B$ of $A$. It easily follows that $B$ is a $2K$-net of $M$. □
The concept of coarse quasi-isometry was introduced by M. Gromov as follows [?]; it is also called *rough isometry* in the context of potential theory [?]. Let $M'$ be another metric space with metric $d'$. Then a *coarse quasi-isometry* between $M$ and $M'$ is a bi-lipschitz bijection between nets of $M$ and $M'$; in this case, $M$ and $M'$ are said to have the same *coarse quasi-isometry type* or to be *coarsely quasi-isometric*. A map $M \to M'$ will be said to *induce* a coarse quasi-isometry when its restriction to some subsets of $M$ and $M'$ is a coarse quasi-isometry. A coarse quasi-isometry between $M$ and itself will be called a *coarse quasi-sometric transformation* of $M$. For some $K > 0$ and $C \ge 1$, the pair $(K, C)$ is said to be a *coarse distortion* of a coarse quasi-isometry if it is a bi-lipschitz bijection between $K$-nets with bi-lipschitz distortion $C$. A family of *equi-coarse quasi-isometries* is a collection of coarse quasi-isometries that have a common coarse distortion.

Two coarse quasi-isometries between $M$ and $M'$, $f: A \to A'$ and $g: B \to B'$, are said to be *close* if there are some $r, s > 0$ such that, for all $x \in A$ and $y \in B$, we have

$$d(x, B), d(y, A) < r, \\ d(x, y) \le r \implies d'(f(x), g(y)) \le s.$$

It may be also said that these coarse quasi-isometries are $(r, s)$-close.

It is well known that “being coarsely quasi-isometric” is an equivalence relation on metric spaces. Indeed, this is a consequence of the fact that the composite of coarse quasi-isometries makes sense up to closeness. The following analysis of such “coarse composite” will be used later.

**Lemma 3.2.** For $K > 0$, let $A_1$ and $A_2$ be a $K$-nets of a metric space $M$ with metric $d$. Then each $A_i$ contains a $6K$-net $B_i$ of $M$ such that there is a bi-lipschitz bijection $h: B_1 \to B_2$ with bi-lipschitz distortion 5 and so that $d(x, h(x)) \le 2K$ for all $x \in B_1$.

*Proof.* By Lemma 3.1, each $A_i$ contains some $K$-separated $2K$-net $A'_i$ of $M$. For each $x \in A'_1$, there is some point $h(x) \in A'_2$ such that $d(x, h(x)) \le 2K$, where $d$ is the metric of $M$. A map $f: A'_1 \to A'_2$ is defined in this way, and let $B_2$ denote its image. Choose some point $g(y) \in h^{-1}(y)$ for each $y \in B_2$, defining a map $g: B_2 \to A'_1$, and let $B_1$ denote its image. Then the restriction $f: B_1 \to B_2$ is bijective with inverse equal to the restriction $g: B_2 \to B_1$.

For any $z \in M$, there is some $x \in A'_1$ with $d(z, x) \le 2K$. Then $h(x) \in B_2$, $g \circ h(x) \in B_1$, and

$$d(z, h(x)) \le d(z, x) + d(x, h(x)) \le 4K, \\ d(z, g \circ h(x)) \le d(z, h(x)) + d(h(x), g \circ h(x)) \le 6K.$$

It follows that $B_1$ and $B_2$ are $6K$-nets of $M$.

On the other hand, since $B_1$ and $B_2$ are $K$-separated and $f: B_1 \to B_2$ is bijective, we have

$$d(h(x), h(y)) \le d(h(x), x) + d(x, y) + d(y, h(y)) \le d(x, y) + 4K \le 5 d(x, y), \\ d(x, y) \le d(x, h(x)) + d(h(x), h(y)) + d(h(y), y) \le d(h(x), h(y)) + 4K \le 5 d(h(x), h(y))$$

for different points $x, y \in B_1$.

□
**Proposition 3.3.** Let $M, M'$ and $M''$ be metric spaces with metrics $d, d'$ and $d''$. Let $f: A \to A'_1$ be a coarse quasi-isometry between $M$ and $M'$, and let $f': A'_2 \to A''$ be a coarse quasi-isometry between $M'$ and $M''$, both of them with coarse distortion $(K, C)$. Then there is a coarse quasi-isometry $g: B \to B''$ between $M$ and $M''$ with coarse distortion $(6KC, 5C^2)$ so that $B \subset A, B'' \subset A''$ and $d'(f(x), f'^{-1} \circ g(x)) \le 2K$ for all $x \in B$. Moreover $g$ is the composite of coarse quasi-isometries close to $f$ and $f'$.

*Proof.* By Lemma 3.2, each $A'_i$ contains a $6K$-net $B'_i$ of $M'$ such that there is a bi-lipschitz bijection $h: B'_1 \to B'_2$ with bi-lipschitz distortion 5 so that $d'(x', h(x')) \le 2K$ for all $x' \in B'_1$. It easily follows that $B = f^{-1}(B'_1)$ and $B'' = f'(B'_2)$ are $6KC$-nets of $M'$ and $M''$, respectively, and the composite

$$B \xrightarrow{f} B'_1 \xrightarrow{h} B'_2 \xrightarrow{f'} B''$$

is a bi-lipschitz bijection $g: B \to B''$ with bi-lipschitz distortion $5C^2$. Finally

$$d'(f(x), f'^{-1} \circ g(x)) = d'(f(x), h \circ f(x)) \le 2K$$

for each $x \in B$.

It only remains to prove that the coarse quasi-isometries $f: B \to B'_1$ and $f' \circ h: B'_1 \to B''$ are coarsely equivalent to $f: A \to A'_1$ and $f' : A'_2 \to A''$, respectively.

Take any $x \in A$ and $y \in B$. Then $d(x, B) \le 6CK$ because $B$ is a $6CK$-net in $M$, and $d(y, A) = 0$ because $B \subset A$. Moreover $d(f(x), f(y)) \le 6C^2K$ if $d(x, y) \le 6CK$, obtaining that $f: B \to B'_1$ is $(6CK, 6C^2K)$-coarsely equivalent to $f: A \to A'_1$.

Now let $x' \in A'_2$ and $y' \in B'_1$. Then $d'(x', B'_1) \le 6K$ because $B'_1$ is a $6K$-net in $M'$, and $d'(y', A'_2) \le K$ because $A'_2$ is a $K$-net in $M'$. Furthermore

$$d''(f'(x'), f' \circ h(y')) \le C d'(x', h(y')) \le C d'(x', y') + C d'(y', h(y')) \le 8CK$$

if $d'(x', y') \le 6K$, yielding that $f' \circ h: B'_1 \to B''$ is $(6K, 8CK)$-close to $f'' : A'_2 \to A''$. $\square$

In Proposition 3.3, the map $g$ represents the “corse composite” of $f$ and $f'$. More precisely, the coarse equivalence classes of coarse isometries between metric spaces form a category of isomorphisms ***. This is indeed the subcategory of isomorphisms of the following larger category. For any set $S$ and a metric space $M$, with metric $d$, two maps $f, g: S \to M$ are said to be *close*<fnref n="1" /> when there is some $R > 0$ such that $d(f(x), g(x)) < R$ for all $x \in S$; it may be also said that these maps are *R-close*. If $M'$ is another metric space, with metric $d'$, a (possibly non-continuous) map $f: M \to M'$ is said to be *large scale lipschitz* [?] if there are constants $\lambda \ge 1$ and $c > 0$ such that

$$d'(f(x), f(y)) \le \lambda d(x, y) + c$$

for all $x, y \in M$; in this case, the pair $(\lambda, c)$ will be called a *large scale lipschitz distortion* of $f$. The map $f$ is said to be *large scale bi-lipschitz* if there are constants $\lambda \ge 1$ and $c > 0$ such that

$$\frac{1}{\lambda} d(x, y) - c \le d'(f(x), f(y)) \le \lambda d(x, y) + c$$

<footnote n="1">¹This term is used in [?]. Other terms used to indicate the same property are *closely equivalent* [?], *parallel* [?], *bornotopic* [?], and *uniformly close* [?].</footnote>
for all $x, y \in M$; in this case, the pair $(\lambda, c)$ will be called a *large scale bi-lipschitz distortion* of $f$. The map $f$ will be called a *large scale lipschitz equivalence* if it is large scale lipschitz and there is another large scale lipschitz $g : M' \to M$ so that $g \circ f$ and $f \circ g$ are close to the identity maps on $M$ and $M'$, respectively. In this case, if $(\lambda, c)$ is a large scale lipschitz distortion of $f$ and $g$, and $g \circ f$ and $f \circ g$ are $R$-close to the identity maps for some $R > 0$, then $(\lambda, c, R)$ will be called a *large scale lipschitz equivalence distortion* of $f$. A large scale lipschitz equivalence is easily seen to be large scale bi-lipschitz.

It is well known that two metric spaces are coarsely quasi-isometric if and only if they are isomorphic in the category of metric spaces and closeness equivalence classes of large scale lipschitz maps; this is part of the content of the following two results, where the constants involved are specially analyzed.

**Proposition 3.4.** Let $f : A \to A'$ be any coarse quasi-isometry between $M$ and $M'$ with coarse distortion $(K, C)$. Then $f$ is induced by a large scale lipschitz equivalence $\phi : M \to M'$ with large scale lipschitz equivalence distortion $(C, 2CK, K)$.

*Proof.* For each $x \in M$ and $x' \in M'$, choose points $h(x) \in A$ and $h'(x) \in A'$ such that $d(x, h(x)) \le K$ and $d'(x', h'(x')) \le K$. Moreover assume that $h(x) = x$ for all $x \in A$, and that $h'(x') = x'$ for all $x' \in A'$. Then $f$ and $f^{-1}$ are respectively induced by the maps $\phi = f \circ h : M \to M'$ and $\psi = f^{-1} \circ h' : M' \to M$. For all $x, y \in M$, we have

$$ 
\begin{aligned}
d'(\phi(x), \phi(y)) &\le C d(h(x), h(y)) \\
&\le C d(h(x), x) + C d(x, y) + C d(y, h(y)) \\
&\le C d(x, y) + 2CK;
\end{aligned}
$$

furthermore

$$ d(x, \psi \circ \phi(x)) = d(x, h(x)) \le K $$

We similarly get

$$ 
\begin{aligned}
d(\psi(x'), \psi(y')) &\le C d'(x', y') + 2CK, \\
d(x', \phi \circ \psi(x')) &\le K
\end{aligned}
$$

for all $x', y' \in M'$, and the result follows. $\square$

**Proposition 3.5.** Let $\phi : M \to M'$ be large scale lipschitz equivalence with large scale lipschitz equivalence distortion $(\lambda, c, R)$. Then, for each $\varepsilon > 0$, the map $\phi$ induces a coarse quasi-isometry $f : A \to A'$ between $M$ and $M'$ with coarse distortion

$$ \left( R + 2\lambda R + \lambda c + \lambda \varepsilon, \lambda + \frac{2R+c}{\varepsilon} \right). $$

*Proof.* Let $\psi : M' \to M$ be large scale lipschitz maps with large scale lipschitz distortion $(\lambda, c)$ such that $\psi \circ \phi$ and $\phi \circ \psi$ are $R$-close to the identity maps on $M$ and $M'$, respectively.
By Lemma 3.1, there is a $(2R+c+\varepsilon)$-separated $(2R+c+\varepsilon)$-net A of M. Let $A' = \phi(A)$, and let $f: A \to A'$ denote the restriction of $\phi$. For all $x, y \in M$, the inequality

$$d(x, y) \le d(x, \psi \circ \phi(x)) + d(\psi \circ \phi(x), \psi \circ \phi(y)) + d(\psi \circ \phi(y), y)$$

yields

$$(3.1) \qquad d(x, y) \le \lambda d'(\phi(x), \phi(y)) + 2R + c.$$

In particular, we get $d(x, y) \le 2R + c$ if $\phi(x) = \phi(y)$. Therefore $f$ is bijective because A is $(2R + c)$-separated.

For any $x' \in M'$, there is some $x \in A$ such that $d(x, \psi(x')) \le 2R + c + \varepsilon$. Then

$$\begin{aligned} d'(x', \phi(x)) &\le d'(x', \phi \circ \psi(x')) + d'(\phi \circ \psi(x'), \phi(x)) \\ &\le R + \lambda d(\psi(x'), x) + c \\ &\le R + 2\lambda R + \lambda c + \lambda \varepsilon. \end{aligned}$$

So $A'$ is a $(R + 2\lambda R + \lambda c + \lambda \varepsilon)$-net of $M'$.

For different points $x, y \in A$, we have

$$d'(f(x), f(y)) \le \lambda d(x, y) + c \le \left( \lambda + \frac{c}{2R + c + \varepsilon} \right) d(x, y)$$

since A is $(2R+c+\varepsilon)$-separated. By the same reason and (3.1), it follows that $d'(f(x), f(y)) > \varepsilon$. Hence

$$d(x, y) \le \lambda d'(f(x), f(y)) + 2R + c \le \left( \lambda + \frac{2R + c}{\varepsilon} \right) d'(f(x), f(y))$$

again by (3.1), which finishes the proof. □

<figure><img src="image_3.png" /></figure>

It is well known that any coarse quasi-isometry can be transformed into another coarse quasi-isometry which is close to the original one and maps any given point to any other given point. The following result keeps track of the constants involved in such a transformation of coarse quasi-isometries.

**Proposition 3.6.** Let $f: A \to A'$ be a coarse quasi-isometry between metric spaces M and M' with coarse distortion $(K, C)$. Suppose that

$$f(B_M(x, R) \cap A) \cap B_{M'}(x', R) \ne \emptyset$$

for some $R > 0$ and some points $x \in M$ and $x' \in M'$. Then there is a coarse quasi-isometry $g: B \to B'$ with coarse distortion $(R + 2CR + K, 2C + 1)$, which is $(r, s)$-close to $f$ for $r = R + 2CR + K$ and $s = 2CR + 2C^2R + CK + R$, and such that $x \in B, x' \in B'$ and $f(x) = x'$.
Proof. By hypothesis, there is some $y_0 \in A$ such that $d(x, y_0) < R$ and $d'(x', f(y_0)) < R$. So

$$B_{M'}(x', R) \subset B_{M'}(f(y_0), 2R),$$

yielding

$$ \begin{aligned} A' \cap B_{M'}(x', R) &\subset A' \cap B_{M'}(f(y_0), 2R) \\ &\subset f(A \cap B_{M}(y_0, 2CR)) \\ &\subset f(A \cap B_{M}(x, 2CR + R)) \end{aligned} $$

since $C$ is a bi-lipschitz distortion of $f$. By the same reason,

$$ \begin{aligned} f(A \cap B_{M}(x, 2CR + R)) &\subset f(A \cap B_{M}(y_0, 2CR + 2R)) \\ &\subset A' \cap B_{M'}(f(y_0), 2C^2R + 2CR) \\ &\subset A' \cap B_{M'}(x', 2C^2R + 2CR + R). \end{aligned} $$

Now let

$$ \begin{aligned} B &= \{x\} \cup (A \setminus B_{M}(x, 2CR + R)), \\ B' &= \{x'\} \cup (B \setminus f(A \cap B_{M}(x, 2CR + R))). \end{aligned} $$

It follows from the above inclusions that

$$(3.2) \qquad B' \cap B_{M'}(x', R) = \{x'\},$$

$$(3.3) \qquad A' \setminus B_{M'}(x', 2C^2R + 2CR + R) \subset B,$$

To show that $B$ is a $(R + 2CR + K)$-net in $M$, take any $y \in M$. It can be assumed that $d(x, y) > 2CR + R + K$ because $x \in B$. Since $A$ is a $K$-net in $M$, there is some $z \in A$ such that $d(y, z) \le K$. Then

$$d(x, z) \ge d(y, x) - d(y, z) > 2CR + R,$$

yielding that $z \in B$, and thus $d(y, B) \le K$.

Now, to prove that $B'$ is a $(2C^2R + 2CR + R)$-net in $M'$, take any $y' \in M'$. Again, it can be assumed that $d(x', y') > 2C^2R + 2CR + R$ because $x' \in B'$. Since $A'$ is a $K$-net in $M'$, there is some $z' \in A'$ such that $d(y', z') \le K$. Then

$$d(x', z') \ge d(y', x') - d(y', z') > 2C^2R + 2CR + R,$$

yielding that $z' \in B'$ by (3.3), and thus $d(y', B') \le K$.
Let $g: B \to B'$ be the bijection defined by $g(z) = f(z)$ if $z \neq x$, and $g(x) = x'$. For any $y \in B \setminus \{x\}$, we have $d(x, y) > R$ and $d'(x', f(y)) > R$ by (3.2). Therefore

$$
\begin{align*}
d'(g(x), g(y)) &= d'(x', f(y)) \\
&\le d'(x', f(y_0)) + d'(f(y_0), f(y)) \\
&< R + C d(y_0, y) \\
&\le R + C d(y_0, x) + C d(x, y) \\
&< R + CR + C d(x, y) \\
&< (2C + 1) d(x, y), \\
d(x, z) &\le d(x, y) + d(y, z) \\
&< R + C d'(f(y), f(z)) \\
&\le R + C d'(f(y), x') + C d'(x', f(z)) \\
&< R + CR + C d'(x', f(z)) \\
&< (2C + 1) d'(x', f(z)) \\
&= (2C + 1) d'(g(x), g(z)).
\end{align*}
$$

Then $g$ is bi-lipschitz with bi-lipschitz distortion $2C + 1$ because it is equal to $f$ on $B \setminus \{x\}$.

Fix arbitrary points $y \in A$ and $z \in B$. We have $d(y, B) \le R + 2CR + K$ because $B$ a $(R + 2CR + K)$-net of $M$, and $d(z, A) \le K$ because $A$ is a $K$-net. Now assume that $d(y, z) \le R + 2CR + K$. If $z \ne x$, then $z \in A$ and $g(z) = f(z)$, yielding

$$d'(f(y), g(z)) \le C d(y, z) \le CR + 2C^2R + CK.$$

If $z = x$, then

$$
\begin{align*}
d'(f(y), g(z)) &= d'(f(y), x') \\
&\le d'(f(y), f(y_0)) + d'(f(y_0), x') \\
&< C d(y, y_0) + R \\
&\le C d(y, z) + C d(z, y_0) + R \\
&< 2CR + 2C^2R + CK + R.
\end{align*}
$$

So $g$ is $(r, s)$-close to $f$ with the constants $r, s$ of the statement.

<figure><img src="image_4.png" /></figure>

The following result will be useful to produce coarse quasi-isometries. It is a coarse quasi-isometry version of the Ascoli-Arzela theorem.

**Proposition 3.7.** Let $M, M'$ be metric spaces with metrics $d, d'$. For $K > 0$ and $C \ge 1$, suppose that:

* there are exhausting increasing sequences of finite subsets $F_n$ and $F'_n$ of $M$ and $M'$, respectively;
* for each $n$, there is a coarse quasi-isometry $f_n$ between $(F_n, d)$ and $(F'_n, d')$ with coarse distortion $(K, C)$;

* $f(F_m \cap \text{dom } f_n) = F'_m \cap \text{im } f_n$ if $m < n$.

Then there is a coarse quasi-isometry $g$ between $M$ and $M'$ with coarse distortion $(K, C)$, which is the combination of restrictions $f_{n_m} : F_m \cap \text{dom } f_{n_m} \to F'_m \cap \text{im } f_{n_m}$ for some subsequence of maps $f_{n_m}$.

*Proof.* For each $m$, let $\mathcal{C}_m$ be the set of restrictions $f_n : F_m \cap \text{dom } f_n \to F'_m \cap \text{im } f_n$ for all $n \ge m$. Define a graph structure on $\mathcal{C} = \bigcup_m \mathcal{C}_m$ by adding an edge between each $f \in \mathcal{C}_{m+1}$ and its restriction $F_m \cap \text{dom } f \to F'_m \cap \text{im } f$, which is well defined and belongs to $\mathcal{C}_m$; such a $\mathcal{C}$ is an infinite tree. Each $\mathcal{C}_m$ is finite because so are the sets $F_m$ and $F'_m$, and thus $\mathcal{C}$ is locally finite (each vertex meets a finite number of edges). Therefore $\mathcal{C}$ contains an infinite ray with vertices $g_m \in \mathcal{C}_m$; every $g_m$ is a restriction $f_{n_m} : F_m \cap \text{dom } f_{n_m} \to F'_m \cap \text{im } f_{n_m}$. All maps $g_m$ can be combined to define a map $g : \bigcup_m \text{dom } g_m \to \bigcup_m \text{im } g_m$. Since the sequences $F_n$ and $F'_n$ are exhausting in $M$ and $M'$, this $g$ is easily seen to be a coarse quasi-isometry between $M$ and $M'$ with coarse distortion $(K, C)$. □

For a metric space $M$ with metric $d$, a subset $X$ of $M$, and some $r > 0$, recall that the (positive closed) penumbra of $X$ of radius $r$ in $M$ is the set

$$ \text{Pen}(X, r) = \{x \in M \mid d(x, X) \le r\}; $$

in particular, the penumbra of a point is a closed ball. So a subset $A \subset X$ is a $K$-net if and only if $\text{Pen}(A, K) = X$.

Recall that the set of vertices $V$ of any connected graph $M$ has a canonical metric: the distance between two points $x, y \in V$ is the minimum number of consecutive edges of $M$ needed to join $x$ and $y$. In this case, it is easy to check that

$$ (3.4) \qquad \text{Pen}(X, r) = \text{Pen}(\text{Pen}(X, r - K), K) $$

for $X \subset V$ and $0 < K \le r$. The following result will be useful to apply Proposition 3.7.

**Lemma 3.8.** Let $V$ be the set of vertices of a connected graph with the corresponding metric, and let $A$ be a $K$-net in $M$ for some $K > 0$. Then, for any subset $X \subset M$ and any $r \ge K$, the set $\text{Pen}(X, r) \cap A$ is a $2K$-net in $\text{Pen}(X, r)$.

*Proof.* By (3.4), for any $x \in \text{Pen}(X, r)$, there is some $y \in \text{Pen}(X, r - K)$ with $d(x, y) \le K$, and there is some $z \in A$ with $d(y, z) \le K$ since $A$ is a $K$-net. So $z \in \text{Pen}(X, r) \cap A$ by (3.4), and $d(x, z) \le 2K$ by the triangle inequality. □

## 3.2 Coarse structures

The notion of coarse structure was introduced in [?] and [?]. It is an abstraction of the closeness relation on maps of any set to a metric space, whose basic definitions and results are recalled here. A coarse structure on a set $X$ is defined by giving, for each set $S$, an equivalence relation (called “being close”) on the set of maps $S \to X$ such that:
(i) if $p, q : S \to X$ are close and $h : S' \to S$ is any map, then $p \circ h$ and $q \circ h$ are close;

(ii) if $S = S' \cup S''$ and if $p, q : S \to X$ are maps whose restrictions to both $S'$ and $S''$ are close, then $p$ and $q$ are close; and

(iii) all constant maps $S \to X$ are close to each other.

A set equipped with a coarse structure is called a *coarse space*.

Let $X$ be a coarse space. A subset $E \subset X \times X$ is called *controlled*<fnref n="2" /> if the restriction to $E$ of the two factor projections $X \times X \to X$ are close. The coarse structure of $X$ is determined by its controlled sets: two maps $p, q : S \to X$ are close if and only if the image of $(p, q) : S \to X \times X$ is controlled. Thus a coarse structure can be also defined in terms of its controlled sets [?], [?]. A subset $B \subset X$ is called *bounded* if $B \times B$ is controlled, which holds if and only if the inclusion map $B \hookrightarrow X$ is close to a constant map. With more generality, a collection $\mathcal{U}$ of subsets of $X$ is said to be *uniformly bounded* if $\bigcup_{U \in \mathcal{U}} U \times U$ is controlled. The coarse space $X$ is called *separable* if it has a countable uniformly bounded cover.

A function $f : X \to X'$ between coarse spaces is called a *coarse map* if

(i) whenever $p, q : S \to X$ are close maps, the composites $f \circ p, f \circ q : S \to X'$ are close maps; and

(ii) if $B$ is a bounded subset of $X'$, then $f^{-1}(B)$ is bounded in $X$.

The coarse spaces $X$ and $X'$ are said to be *coarsely equivalent* if there exist coarse maps $f : X \to X'$ and $g : X' \to X$ such that $g \circ f$ and $f \circ g$ are close to the identity maps on $X$ and $X'$, respectively. In this case, the map $f$ is called a *coarse equivalence*. The category of coarse spaces and closeness equivalence classes of coarse maps is called the *coarse category*.

If $X$ is also equipped with a locally compact topology, the coarse structure of $X$ is said to be *proper* when

(i) $X$ has a uniformly bounded open cover; and

(ii) every bounded subset of $X$ has compact closure.

In this case, $X$ will be just called a *proper coarse space*. Note that bounded subsets of a proper coarse space are just the ones with compact closure.

For our purposes, the most important example of coarse structure is given by any metric space $M$, with metric $d$. Recall from Section 3.1 that, for any set $S$, two maps $f, g : S \to M$ are said to be *close* when $\sup\{d(f(s), g(s)) \mid s \in S\} < \infty$. This closeness relation defines a coarse structure on $M$, which is called its *metric*<fnref n="3" /> coarse structure. The terms *metric closeness* and *metric controlled set* can be used in this case. This coarse structure is proper if and only if the metric space $M$ is proper in the sense that its closed balls are compact. In the case of metric coarse structures, the above abstract coarse notions have their usual meanings for metric spaces.

<footnote n="2">²Controlled sets are called *entourages* in [?].</footnote>

<footnote n="3">³This term is taken from [?]. It is also called *bounded coarse structure* in [?]</footnote>
With more generality, using the terminology from [?], a coarse distance (or coarse metric) on a set $M$ is a symmetric map $d: M \times M \to [0, \infty)$ satisfying the triangle inequality; in this case, $(M, d)$ is called a coarse metric space. Any coarse distance defines a coarse structure in the same way as a metric does, and will be also called a *metric coarse structure*. In this section and the above one, all notions and properties are given for metric spaces for simplicity, but they have obvious versions for coarse metric spaces.

For any compact foliated space $X$ (without any smoothness condition), it will be seen that any finite atlas $\mathcal{U}$ induces a coarse metric $d_U$ on its leaves by using the length of paths of plaques; such $d_U$ depends on the choice of $\mathcal{U}$, but the corresponding metric coarse structure depends only on $X$.

If $M$ and $M'$ are metric spaces, with distance functions $d$ and $d'$, the two conditions on a map $f: M \to M'$ to be coarse can be written as follows:

(i) (Uniform expansiveness$^4$.) For each $R > 0$ there is some $S > 0$ such that

$$d(x, y) \le R \implies d'(f(x), f(y)) \le S$$

for all $x, y \in M$.

(ii) (Metric properness$^5$.) For each bounded subset $B \subset M'$, the inverse image $f^{-1}(B)$ is bounded in $M$.

The last property has a uniform version: a map $f: M \to M'$ is said to be *uniformly metric proper*$^6$ if for each $R > 0$ there is some $S > 0$ so that

$$d'(f(x), f(y)) < R \implies d(x, y) < S$$

for all $x, y \in M$. By using uniform metric properness instead of metric properness, we get what is called the *rough category*. More precisely, a map between metric spaces, $f: M \to M'$, is called a *rough map*$^7$ if it is uniformly expansive and uniformly metric proper; if moreover there is a rough map $g: M' \to M$ so that the compositions $g \circ f$ and $f \circ g$ are respectively close to the identity maps on $M$ and $M'$, then $f$ is called a *rough equivalence*; in this case, $M$ and $M'$ are said to be *roughly equivalent*$^8$. Thus rough equivalences are the maps which induce isomorphisms in the rough category. There are interesting differences between the rough category and the coarse category of metric spaces [?], but the following shows that they have the same isomorphisms.

**Proposition 3.9.** *Any coarse equivalence between metric spaces is uniformly metric proper. Moreover the definition of uniformly metric properness is satisfied with constants that depend only on the constants involved in the definition of coarse equivalence.*

<footnote>$^4$This term comes from [?]. Other terms used to denote the same property are *uniformly bornologous* [?] and *coarsely lipschitz* [?].</footnote>

<footnote>$^5$This term is used in [?].</footnote>

<footnote>$^6$This term is used in [?]. Another term used to denote the same property is *effectively proper* [?] ***.</footnote>

<footnote>$^7$***</footnote>

<footnote>$^8$The term *uniform closeness* is used in [?] to indicate this equivalence between metric spaces</footnote>
Proof. Let $f: M \to M'$ and $g: M' \to M$ be coarse maps so that $g \circ f$ and $f \circ g$ are $r$-close to the identity maps on $M$ and $M'$ for some $r > 0$. Then, for any $R > 0$ there is some $S > 0$ such that

$$d'(x', y') < R \implies d(g(x'), g(y')) < S$$

for all $x', y' \in M'$. It follows that, if $d'(f(x), f(y)) < R$ for some $x, y \in M$, then

$$d(x, y) \le d(x, g \circ f(x)) + d(g \circ f(x), g \circ f(y)) + d(g \circ f(y), y) < S + 2r. \quad \square$$

There is no way to define “equi-coarse maps” or “equi-corse equivalences” between arbitrary coarse spaces, but we can define the following in the case of metric coarse structures. A collection of maps $f_i: M_i \to M'_i, i \in \Lambda$, is said to be a family of:

* equi-uniformly expansive maps if they satisfy the condition of uniform expansiveness involving the same constants;

* equi-uniformly metric proper maps if they satisfy the condition of uniformly metric properness involving the same constants;

* equi-rough maps if they are equi-uniformly expansive and equi-uniformly metric proper; and

* equi-rough equivalences if they are equi-rough, and there is another collection of equi-rough maps $g_i: M'_i \to M_i, i \in \Lambda$, and there is some $r > 0$ so that the composites $g_i \circ f_i$ and $f_i \circ g_i$ are $r$-close to the identity maps on $M_i$ and $M'_i$, respectively, for all $i \in \Lambda$.

According to Proposition 3.9, a collection of equi-rough equivalences can be also properly called a family of equi-coarse equivalences.

Now, recall that a metric space $M$, with metric $d$, is said to be convex if, for each pair of points $x, y \in M$, there is some $z \in M$ such that $d(x, y) = d(x, z) + d(y, z)$. By iterating the convexity condition we get that, for each $x, y \in M$ and any $\varepsilon > 0$, there is some finite sequence of points $x = x_0, \dots, x_n = y$ in $M$ so that $d(z_{k-1}, z_k) < \varepsilon$ for $k = 1, \dots, n$, and

$$\sum_{k=1}^{n} d(x_{k-1}, x_k) = d(x, y).$$

The following is kind of a coarse version of this last property.

**Definition 3.10.** The metric space $M$ is said to be coarsely convex if there are constants $a, b, c > 0$ such that, for each $x, y \in M$, there is some finite sequence of points $x = x_0, \dots, x_n = y$ in $M$ such that $d(x_{k-1}, x_k) \le c$ for all $k \in \{1, \dots, n\}$, and

$$\sum_{k=1}^{n} d(x_{k-1}, x_k) \le a d(x, y) + b.$$

A family of metric spaces is said to be equi-coarsely convex if all of them satisfy the condition of coarse convexity involving the same constants.
A typical example of a coarsely convex metric space, which is not convex, is the set $V$ of vertices of a connected graph $M$ with the metric $d_V$ induced by $M$ (Section 3.1). Note that the coarse convexity condition is satisfied in this case with the constants $a = b = c = 1$. This metric on $V$ is the restriction of a metric on $M$ that can be defined as follows. Choose any metric $d_e$ on each edge $e$ of $M$ so that $e$ is isometric to a unit interval. Then the distance between two points $x, y \in M$ is the minimum of the sums of the form

$$d_e(x, v) + d_V(v, w) + d_f(w, y),$$

where $x, y$ lie in edges $e, f$, and $v, w$ are vertexes of $e, f$, respectively. Observe that $M$ is convex and $V$ is a 1/2-net in $M$. With more generality, any net of a convex metric space is coarsely convex, which is a particular case of the following result.

**Theorem 3.11.** A metric space $M$ is coarsely convex if and only if there exists a coarse quasi-isometry $f: A \to A'$ between $M$ and some convex metric space $M'$. In this case, the coarsely quasi-isometric distortion of $f$ depends only on the constants involved in the condition of coarse convexity satisfied by $M$, and viceversa; equivalently, a family of metric spaces is equi-coarsely convex if and only if they are equi-corsely quasi-isometric to convex metric spaces.

*Proof.* Let $d$ be the metric of $M$. Suppose that there is a coarse quasi-isometry $f: A \to A'$, with coarse distortion $(K, C)$, between $M$ and a convex metric space $M'$, with metric $d'$.

For all $x, y \in M$, there are some $\bar{x}, \bar{y} \in A$ with $d(x, \bar{x}), d(y, \bar{y}) \le K$. Then there is some finite sequence $f(\bar{x}) = x'_0, \dots, x'_n = f(\bar{y})$ in $M'$ such that $d'(x'_{k-1}, x'_k) < 1$ for all $k \in \{1, \dots, n\}$ and

$$\sum_{k=1}^{n} d'(x'_{k-1}, x'_{k}) = d'(f(\bar{x}), f(\bar{y})).$$

Moreover, we can assume that this is one of the shortest sequences satisfying this condition. If we had $d'(x'_{k-1}, x'_k) < 1/2$ and $d'(x'_k, x'_{k+1}) < 1/2$ for some $k$, then the term $x'_k$ could be removed from the sequence, contradicting its minimality. It follows that $d'(x'_{k-1}, x'_k) \ge 1/2$ for at least $[n/2]$ indices $k$. So

$$(n-1)/4 \le [n/2]/2 \le \sum_{k=1}^{n} d'(x'_{k-1}, x'_{k}) = d'(f(\bar{x}), f(\bar{y})).$$

yielding

$$(3.5) \qquad n \le 4 d'(f(\bar{x}), f(\bar{y})) + 1.$$

For each $k \in \{0, \dots, n\}$, there is some $\bar{x}'_k \in A'$ with $d'(\bar{x}'_k, x'_k) \le K$, and let $\bar{x}_k = f^{-1}(\bar{x}'_k)$; for simplicity, take $\bar{x}'_0 = x'_0$ and $\bar{x}'_n = x'_n$, and thus $\bar{x}_0 = \bar{x}$ and $\bar{x}_n = \bar{y}$. To simplify the
notation, let also $x_0 = x$, $x_n = y$, and $x_k = \bar{x}_k$ for $k \in \{1, \dots, n-1\}$. Then

$$
\begin{aligned}
d(x_{k-1}, x_k) &\le d(x_{k-1}, \bar{x}_{k-1}) + d(\bar{x}_{k-1}, \bar{x}_k) + d(\bar{x}_k, x_k) \\
&\le 2K + C d'(\bar{x}'_{k-1}, \bar{x}'_k) \\
&\le 2K + C (d'(\bar{x}'_{k-1}, x'_{k-1}) + d'(x'_{k-1}, x'_k) + d'(x'_k, \bar{x}'_k)) \\
&\le 2K + 2CK + C
\end{aligned}
$$

for $k \in \{1, \dots, n\}$, and

$$
\begin{aligned}
\sum_{k=1}^{n} d(x_{k-1}, x_k) &\le d(x, \bar{x}) + \sum_{k=1}^{n} d(\bar{x}_{k-1}, \bar{x}_k) + d(\bar{y}, y) \\
&\le 2K + C \sum_{k=1}^{n} d'(\bar{x}'_{k-1}, \bar{x}'_k) \\
&\le 2K + C \sum_{k=1}^{n} (d'(\bar{x}'_{k-1}, x'_{k-1}) + d'(x'_{k-1}, x'_k) + d'(x'_k, \bar{x}'_k)) \\
&\le 2K + 2CKn + C \sum_{k=1}^{n} d'(x'_{k-1}, x'_k) \\
&\le 2K + 2CK (4 d'(f(\bar{x}), f(\bar{y})) + 1) + C d'(f(\bar{x}), f(\bar{y})) \\
&\le 2K + 2CK + (8CK + C)C d(\bar{x}, \bar{y}) \\
&\le 2K + 2CK + (8CK + C)C (d(\bar{x}, x) + d(x, y) + d(y, \bar{y})) \\
&\le 2K + 2CK + 2(8CK + C)CK + (8CK + C)C d(x, y),
\end{aligned}
$$

where (3.5) was used in the fifth inequality. Thus the condition of Definition 3.10 is satisfied with a, b and c depending only on K and C as desired.

Now assume that $M$ satisfies the coarse convexity condition (Definition 3.10) with constants $a, b$ and $c$. By Lemma 3.1, there is a $c$-separating $c$-net $A$ in $M$. By attaching an edge to any pair of points $x, y \in A$ with $d(x, y) \le 3c$, we get a graph $M'$ whose set of vertices is $A$. For any $x, y \in A$, there is a finite sequence $x = x_0, \dots, x_n = y$ in $M$ with $d(x_{k-1}, x_k) \le c$ for all $k \in \{1, \dots, n\}$, and

$$
\sum_{k=1}^{n} d(x_{k-1}, x_k) \le a d(x, y) + b.
$$

For each $k$, take some $\bar{x}_k \in A$ with $d(x_k, \bar{x}_k) \le c$; in particular, take $\bar{x}_0 = x$ and $\bar{x}_n = y$. Then there is an edge between each $\bar{x}_{k-1}$ and $\bar{x}_k$ because

$$
d(\bar{x}_{k-1}, \bar{x}_k) \le d(\bar{x}_{k-1}, x_{k-1}) + d(x_{k-1}, x_k) + d(x_k, \bar{x}_k) \le 3c.
$$

So $M'$ is a connected graph. Let $d'$ denote the convex metric on $M'$, defined as above, with the choice of a metric on each of its edges so that it becomes isometric to the unit interval. Since $A$ is a 1-net in $M'$, it only remains to check that the identity map $(A, d) \to (A, d')$ is
bi-lipschitz with bi-lipschitz distortion depending only on a, b and c. Fix any pair of different points $x, y \in A$, and take a sequence $x = \bar{x}_0, \dots, \bar{x}_n = y$ as above; by removing some points of this sequence if necessary, we can assume that $\bar{x}_{k-1} \neq \bar{x}_k$ for all $k$. Since there is an edge between each $\bar{x}_{k-1}$ and $\bar{x}_k$, it follows that $d'(x, y) \le n$. But

$$3cn \le \sum_{k=1}^{n} d(\bar{x}_{k-1}, \bar{x}_k) \le a d(x, y) + b \le \frac{ac+b}{c} d(x, y)$$

because A is c-separating, yielding

$$d'(x, y) \le \frac{ac+b}{3c^2} d(x, y).$$

On the other hand, for $d'(x, y) = m$, there is a sequence $x = y_0, \dots, y_m = y$ in A such that there is an edge between each $y_{k-1}$ and $y_k$; thus $d(y_{k-1}, y_k) \le 3c$ for each $k$, yielding

$$d(x, y) \le \sum_{k=1}^{m} d(y_{k-1}, y_k) \le 3cm = 3c d'(x, y). \quad \square$$

**Proposition 3.12.** The following holds:

(i) Any large scale lipschitz map between metric spaces is uniformly expansive; moreover, a family of equi-large scale lipschitz maps between metric spaces is equi-uniformly expansive.

(ii) Any large scale lipschitz equivalence is a rough equivalence; moreover, a family of equi-large scale lipschitz equivalences between metric spaces is a family of equi-rough equivalences.

*Proof.* Let $M$ and $M'$ be metric spaces with metrics $d$ and $d'$, respectively, and let $f: M \to M'$ be a large scale lipschitz map. If $(\lambda, c)$ is a large scale lipschitz distortion of $f$, then $f$ obviously satisfies the definition of uniform expansiveness with $S = \lambda R + c$ for each $R > 0$. This proves property (i) because $S$ depends only on $R$, $\lambda$ and $c$.

Now suppose that moreover $f$ is a large scale lipschitz equivalence. So there is a large scale lipschitz map $g: M' \to M$, whose large scale lipschitz distortion can be assumed to be $(\lambda, c)$ too, such that $g \circ f$ and $f \circ g$ are $r$-close to the identity maps on $M$ and $M'$ for some $r > 0$. Then

$$d(x, y) \le d(x, g \circ f(x)) + d(g \circ f(x), g \circ f(y)) + d(g \circ f(y), y) \\ \le \lambda d'(f(x), f(y)) + 2r.$$

Hence $f$ satisfies the definition of uniform metric properness with $S = \lambda R + 2r$ for each $R > 0$. This proves property (ii) because $S$ depends only on $R$, $\lambda$ and $r$. $\square$
**Example 3.13.** Let $\mathbb{N}^2 = \{n^2 \mid n \in \mathbb{N}\}$ and $\mathbb{N}^3 = \{n^3 \mid n \in \mathbb{N}\}$ with the restriction of the euclidean metric on $\mathbb{R}$. Suppose that $\mathbb{N}^2$ and $\mathbb{N}^3$ are large scale lipschitz equivalent; i.e., there are large scale lipschitz maps $f: \mathbb{N}^2 \to \mathbb{N}^3$ and $g: \mathbb{N}^3 \to \mathbb{N}^2$ with large scale lipschitz distortion $(\lambda, c)$ such that $g \circ f$ and $f \circ g$ are close to identity maps on $\mathbb{N}^2$ and $\mathbb{N}^3$. Let $\sigma, \tau: \mathbb{N} \to \mathbb{N}$ be the maps defined by $f(n^2) = \sigma(n)^3$ and $g(n^3) = \tau(n)^2$. Since $(n+1)^2 - n^2 \to \infty$ and $(n+1)^3 - n^3 \to \infty$ as $n \to \infty$, there is some $a \in \mathbb{N}$ such that $g \circ f(n^2) = n^2$ and $f \circ g(n^3) = n^3$ for all $n \ge a$, yielding $\tau \circ \sigma(n) = \sigma \circ \tau(n) = n$ for $n \ge a$. Assume for a while that there is some integer $b \ge a$ such that $\tau(n) \ge n+a+2$ for all $n \ge b$. Thus

$$ \tau(\{b, b+1, \dots\}) \subset \{b+a+2, b+a+3, \dots\}, $$

yielding

$$ \{a, a+1, \dots, b+a+1\} \subset \tau(\{0, 1, \dots, b-1\}) $$

because $\{a, a+1, \dots\} \subset \tau(\mathbb{N})$. So

$$ b+1 = \#\{a, a+1, \dots, b+a+1\} \le \#\tau(\{0, \dots, b-1\}) \le b, $$

which is a contradiction. Hence there is some sequence $n_k \uparrow \infty$ in $\mathbb{N}$ such that $\tau(n_k) \le n_k + a + 1$ for all $k$. So

$$ |\tau(n_k)^2 - n_k^2| \le (\tau(n_k) + n_k)^2 \le (2n_k + a + 1)^2. $$

We can assume that $n_k \ge a$ for all $k$. Then

$$ \begin{aligned} n_k^3 &= f \circ g(n_k^3) - f \circ g(n_1^3) + n_1^3 \\ &\le \lambda |g(n_k^3) - g(n_1^3)| + c + n_1^3 \\ &= \lambda |\tau(n_k)^2 - \tau(n_1)^2| + c + n_1^3 \\ &\le \lambda (|\tau(n_k)^2 - n_k^2| + n_k^2 - n_1^2 + |n_1^2 - \tau(n_1)^2|) + c + n_1^3 \\ &\le \lambda ((2n_k + a + 1)^2 + n_k^2 - n_1^2 + (2n_1 + a + 1)^2) + c + n_1^3 \\ &= 5\lambda n_k^2 + 4\lambda(a+1)n_k + 3\lambda n_1^2 + 4\lambda(a+1)n_1 + 2\lambda(a+1)^2 + c + n_1^3, \end{aligned} $$

which is a contradiction because $n_k \to \infty$ as $k \to \infty$. Therefore there is no large scale lipschitz equivalence between $\mathbb{N}^2$ and $\mathbb{N}^3$, and thus these spaces are not quasi-isometric. But they are coarsely equivalent; indeed, the map $n^3 \mapsto n^2$ is distance decreasing, and the map $n^2 \to n^3$ is coarse: if $|n^2 - m^2| < R$, then $n+m < R$ also, so $|n^3 - m^3| < S$ with $S = R^3$.

This example shows that the reciprocal of Proposition 3.12-(ii) does not hold in general. But, according to the following results, coarse equivalences coincide with large scale lipschitz equivalences when coarse convexity is assumed.

**Proposition 3.14.** Any uniformly expansive map of a coarsely convex metric space to another metric space is large scale lipschitz; moreover, a family of equi-uniformly expansive maps between metric spaces, whose domains are equi-coarsely convex, is a family of equi-large scale lipschitz maps.
*Proof.* Let $M$ and $M'$ be metric spaces with metrics $d$ and $d'$, respectively, and let $f: M \to M'$ be a uniformly expansive map. Suppose that $M$ satisfies the definition of coarse convexity with constants $a, b, c > 0$. Fix points $x, y \in M$, and let $x = x_0, \dots, x_n = y$ be a minimal sequence between $x$ and $y$ satisfying the coarse convexity condition.

If $d(x_{k-1}, x_k)$ and $d(x_k, x_{k+1})$ are $< c/2$ for some $k \in \{1, \dots, n-1\}$, then $d(x_{k-1}, x_{k+1}) < c$, and thus $x_k$ could be removed from the sequence, which contradicts its minimality. Hence $d(x_{k-1}, x_k) \ge c/2$ for at least $[n/2]$ indices $k$, yielding

$$a d(x, y) + b > \sum_{k=1}^{n} d(x_{k-1}, x_k) \ge [n/2]c/2 \ge \frac{(n-1)c}{4},$$

and thus

$$(3.6) \qquad n < \frac{4a}{c} d(x, y) + \frac{4b}{c} + 1.$$

Since $f$ is uniformly expansive, there is some $S > 0$ such that $d'(f(z), f(z')) < S$ for all $z, z' \in M$ with $d(z, z') < c$. So

$$d'(f(x), f(y)) \le \sum_{k=1}^{n} d'(f(x_{k-1}), f(x_k)) < nS < \frac{4aS}{c} d(x, y) + \frac{4bS}{c} + S$$

by (3.6), and thus $f$ is large scale lipschitz. Moreover the large scale lipschitz distortion of $f$ depends only on $S$, $a$, $b$ and $c$, which shows the second part of the statement. $\square$

**Corollary 3.15.** Any coarse equivalence between coarsely convex metric spaces is a large scale lipschitz equivalence; moreover, a family of equi-coarsely equivalent convex spaces is a family of equi-large scale lipschitz equivalences.

*Proof.* This is elementary by Proposition 3.14. $\square$

**Corollary 3.16.** Two coarsely convex metric spaces are coarsely quasi-isometric if and only if they are coarsely equivalent; moreover, if $M_i$ and $M'_i$, $i \in \Lambda$, are families of equi-corsely convex metric spaces, then all pairs $M_i$ and $M'_i$ are equi-coarsely quasi-isometric if and only they are equi-coarsely equivalent.

*Proof.* This follows from Propositions 3.4, 3.5 and 3.14, and Corollary 3.15. $\square$

The leaves of compact foliated spaces will be endowed with equi-coarsely convex metrics. Then, by Proposition 3.9 and Corollary 3.16, being equi-coarsely equivalent to each other, being equi-roughly equivalent to each other and being equi-coarsely quasi-isometric to each other are equivalent properties for any family of leaves. Therefore neither coarse equivalences nor rough equivalences between leaves will be explicitly considered; it is enough to consider coarse quasi-isometries.
## 3.3 Metric equivalence relations and length structures

The contents of this work deal with metric structures on the leaves of a foliated space or orbits of a pseudogroup, and they fit into the area of metric equivalence relations. Let $R$ be an equivalence relation on a set $X$; the corresponding equivalence class of each $x \in X$ will be denoted by $R(x)$. A distance (or metric) on $R$ is a function $d: R \to [0, \infty)$ such that

* $d(x, y) = 0$ if and only if $x = y$,

* $d(x, y) = d(y, x)$, and

* $d(x, z) \le d(x, y) + d(y, z)$

for all equivalent points $x, y, z$. Thus a metric $d$ on $R$ is just given by a choice of a metric on the equivalence class $R(x)$ of each $x \in X$: the restriction $d: R(x) \times R(x) \to [0, \infty)$. It will be convenient to consider $d$ as defined on the whole of $X \times X$ by setting $d(x, y) = \infty$ when $(x, y) \notin R$. If the “only if” part is removed from the first condition, then $d$ is called a pseudodistance (or pseudometric) on $R$; and if the whole first condition is removed, then it is called a coarse distance (or coarse metric) on $R$ (with the terminology of [?]). If $d$ is a distance on $R$, the triple $(X, R, d)$ is called a metric equivalence relation. The $d$-balls in the equivalence classes form a base of a topology on $X$, which is said to be induced by $d$. The following notation will be used: let

$$B_d(x, r) = \{y \in R(x) \mid d(x, y) < r\}$$

for $x \in X$ and $r > 0$, and let

$$d(A, B) = \inf\{d(x, y) \mid (x, y) \in R \cap (A \times B)\}$$

for $A, B \subset X$ with $R \cap (A \times B) \ne \emptyset$; in particular, set $d(x, B) = d(\{x\}, B)$ when $R(x) \cap B \ne \emptyset$.

An example of metric equivalence relation is given by a length structure, introduced by M. Gromov [?]. A length structure on a set $X$ is defined by a map $\ell: \mathcal{C} = \bigcup_I \mathcal{C}(I) \to [0, \infty)$, where $I$ runs in the family of the compact intervals of $\mathbb{R}$, and each $\mathcal{C}(I)$ is a set of maps $I \to X$ with image in the equivalence classes of $R$, so that the following properties are satisfied:

* The set $\mathcal{C}$ contains all constant maps from compact intervals to $X$, and $\ell(f) = 0$ if and only if $f$ is constant for each $f \in \mathcal{C}$.

* For compact intervals $I \subset J$, the restriction to $I$ of any element of $\mathcal{C}(J)$ is in $\mathcal{C}(I)$.

* For $a \le b \le c$, if $f \in \mathcal{C}([a, b])$ and $g \in \mathcal{C}([b, c])$ with $f(b) = g(b)$, then the combination $h$ of $f$ and $g$ is in $\mathcal{C}([a, c])$, and morever $\ell(h) = \ell(f) + \ell(g)$.

* If $\phi: I \to J$ is a homeomorphism between compact intervals and if $f \in \mathcal{C}(J)$, then $f \circ \phi \in \mathcal{C}(I)$.

* For $a \le b$ and every $f \in \mathcal{C}([a, b])$, the mapping $t \mapsto \ell(f|_{[a,t]})$ is continuous.
Such a set $\mathcal{C}$ induces an equivalence relation $R_{\mathcal{C}}$ on $X$, which is the set of pairs $(x, y) \in X \times X$ such that there is some $f : [a, b] \to X$ in $\mathcal{C}$ with $f(a) = x$ and $f(b) = y$. Moreover a length structure $\ell : \mathcal{C} \to [0, \infty)$ induces a distance $d_{\ell}$ on $R_{\mathcal{C}}$ by setting

$$d_{\ell}(x, y) = \inf\{\ell(f) \mid f \in \mathcal{C}([a, b]), f(a) = x, f(b) = y\}$$

for $(x, y) \in R_{\mathcal{C}}$ any fixed numbers $a < b$. Thus $(X, R_{\mathcal{C}}, d_{\ell})$ is a metric equivalence relation, which will be said to be *induced* by $\ell : \mathcal{C} \to [0, \infty)$. In [?, ?], a metric space induced by a length structure is called a *length space* or *path metric space*; thus, with more generality, any metric equivalence relation induced by a length structure can be called a *length equivalence relation* or *path metric equivalence relation*. Given a path metric space (or path metric equivalence relation) $X$, its length structure will be usually denoted by $\ell_X$, or simply by $\ell$.

Interesting examples and properties of length structures are studied in [?, ?]). For instance, riemannian manifolds and simplicial metric spaces are length spaces; in particular, (geometric realizations of) graphs are length spaces. Also, according to [?, Examples 1.4-(a), p. 2], any metric equivalence relation $(X, R, d)$ induces a length structure on $X$ as follows. For any compact interval $I$ and any continuous map $f : I \to X$, with respect to the topology induced by $d$ on $X$, let

$$\ell_d(f) = \sup \sum_{k=1}^{n} d(f(t_{k-1}), f(t_k)),$$

where $t_0 < t_1 < \dots < t_n$ runs over all partitions of $I$. If $\mathcal{C}_d$ is the set of all such maps $f$ with $\ell_d(f) < \infty$, then $\ell_d : \mathcal{C}_d \to [0, \infty)$ is a length structure, which is called the *metric length structure* of $(X, R, d)$.

For any length structure $\ell : \mathcal{C} \to [0, \infty)$ on $X$, let $\tilde{\ell} : \tilde{\mathcal{C}} \to [0, \infty)$ be the length structure induced by the metric equivalence relation induced by $\ell$; i.e., $\tilde{\mathcal{C}} = \mathcal{C}_{d\ell}$ and $\tilde{\ell} = \ell_{d\ell}$. The arguments of [?, Proposition 1.6, pp. 3–4] show the following.

**Proposition 3.17 (Gromov).** If $\ell : \mathcal{C}(I) \to [0, \infty)$ is lower semicontinuous for every compact interval $I$, where $\mathcal{C}(I)$ is endowed with the compact-open topology with respect to the topology induced by $d_{\ell}$ on $X$, then $\mathcal{C} \subset \tilde{\mathcal{C}}$ and $\ell = \tilde{\ell}|_{\mathcal{C}}$.

For a metric equivalence relation $(X, R, d)$, let $(X, \tilde{R}, \tilde{d})$ be the metric equivalence relation induced by the length structure induced by $(X, R, d)$; i.e., $\tilde{R} = R_{\mathcal{C}_d}$ and $\tilde{d} = d_{\ell_d}$. Obviously, $\tilde{R} \subset R$, and $d \le \tilde{d}$ on $R_{\mathcal{C}_d}$, but equalities do not hold in general [?, Examples 1.4-(b), pp. 2–3]. The proof of [?, Theorem 1.8] shows the following.

**Theorem 3.18 (Gromov).** For any length equivalence relation $(X, R, d)$, each $(x, y) \in R$ and every $\varepsilon > 0$, there is some $z \in R(x)$ such that

$$(3.7) \qquad \max\{d(x, z), d(y, z)\} \le \frac{1}{2} d(x, y) + \varepsilon.$$

If $(X, R, d)$ is a metric equivalence relation with complete equivalence classes and such that, for each $(x, y) \in R$ and every $\varepsilon > 0$, there is some $z \in R(x)$ satisfying (3.7), then $(X, R, d)$ is a length equivalence relation; more precisely, $R = \tilde{R}$ and $d = \tilde{d}$ in this case.
The following corollary follows by iterating the condition of Theorem 3.18.

**Corollary 3.19.** Let $(X, R, d)$ be a length equivalence relation. Then, for each positive integer $n$, every $\varepsilon > 0$ and any $(x, y) \in R$, there is some finite sequence $(x_0, \dots, x_n)$ in $R(x)$ so that $x_0 = x, x_n = y$, and

$$ \max\{d(x_{k-1}, x_k) \mid k \in \{1, \dots, n\}\} \le \frac{1}{n} d(x, y) + \varepsilon. $$

**Definition 3.20.** A metric equivalence relation $(X, R, d)$ is said to be coarsely convex if all of its equivalence classes are equi-coarsely convex (Definition 3.10).

The following notion defines an important class of maps between length spaces [?, page 9], [?, page 22]. Let $f: X \to Y$ be a map between length spaces *** path metric spaces ***. The map $f$ is called an arc-wise isometry if it is lipschitz and satisfies $\ell(f \circ c) = \ell(c)$ for each lipschitz curve $c: I \to X$. The following weaker concepts will be used too. The map $f$ is said to be arc-wise lipschitz if it is lipschitz and there are some $C > 0$ such that $\ell(f \circ c) \le C \ell(c)$ for each lipschitz curve $c: I \to X$; such a $C$ is called an arc-wise lipschitz distortion. The map $f$ is said to be arc-wise bi-lipschitz if it is lipschitz and there is some $C > 0$ such that

$$ \frac{1}{C} \ell(c) \le \ell(f \circ c) \le C \ell(c) $$

for each lipschitz curve $c: I \to X$; now $C$ is called an arc-wise lipschitz bi-distortion. Finally, $f$ is said to be arc-wise large scale bi-lipschitz if it is lipschitz and there are some $\lambda \ge 1$ and $C > 0$ such that

$$ \frac{1}{\lambda} \ell(c) - C \le \ell(f \circ c) \le \lambda \ell(c) + C $$

for each lipschitz curve $c: I \to X$; now $(\lambda, C)$ is called an arc-wise large scale lipschitz bi-distortion. All of these properties are preserved by compositions. ***

## 3.4 Coarse quasi-isometry type of orbits

A typical example of a metric equivalence relation is given by a set $E$ of generators of a pseudogroup $\mathcal{H}$ of local transformations of a space $Z$. Assume that $E$ is symmetric for the sake of simplicity. For each $h \in \mathcal{H}$ and each $x \in \text{dom } h$, let $|h|_{E,x}$ be defined as follows. If $h$ is the identity around $x$, set $|h|_{E,x} = 0$. Otherwise, $|h|_{E,x}$ is the minimal positive integer $k$ such that $h = h_k \circ \dots \circ h_1$ around $x$ for some $h_1, \dots, h_k \in E$. Let $R_{\mathcal{H}} \subset Z \times Z$ denote the equivalence relation induced by $\mathcal{H}$ (whose equivalence classes are the orbits). Then, for $(x, y) \in R_{\mathcal{H}}$, let

$$ d_E(x, y) = \min\{|h|_{E,x} \mid h \in \mathcal{H}, x \in \text{dom } h, h(x) = y\}. $$

In this way, $E$ induces a distance $d_E: R_{\mathcal{H}} \to [0, \infty)$, and thus $(Z, R_{\mathcal{H}}, d_E)$ is a metric equivalence relation. Note that $d_E$ induces the discrete topology on $Z$.
Each orbit of $\mathcal{H}$ has a graph structure induced by the symmetric set of generators $E$: there is an edge between two points $x, y$ in the same orbit just when $x$ is in the domain of some $h \in E$ and $h(x) = y$. Then $E$ is finite if and only if this graph is of *finite type* (there is a uniform upper bound on the number of edges that meet at every vertex). Observe that the restriction of this graph structure to each orbit $\mathcal{O}$ is connected and induces the restriction of the metric $d_E$ to $\mathcal{O}$.

The above construction is a generalization of the well known right invariant metric on a group induced by a set of generators. But, unlike the case of groups, the coarsely quasi-isometry type of the restriction of $d_E$ to each orbit may depend on the choice of $E$, even if $E$ is finite. This is due to the fact that, besides composition and inversion, the restriction and combination of maps are also used to generate $\mathcal{H}$. Moreover an equivalence of pseudogroups may not preserve the coarsely quasi-isometry type of the orbits for any choice of generators, as can be seen in the following simple example.

**Example 3.21.** Let $\mathcal{H}$ be the pseudogroup on $\mathbb{R}$ generated by the action of $\mathbb{Z}$ by translations, and let $\mathcal{G}$ be the restriction of $\mathcal{H}$ to some open interval $U \subset \mathbb{R}$. If $U$ is of length > 1, then it meets every $\mathcal{H}$-orbit, and thus $\mathcal{H}$ is equivalent to $\mathcal{G}$. But the $\mathcal{H}$-orbits are infinite, while each $\mathcal{G}$-orbit is finite if $U$ is of bounded length. So, for the metrics induced by any choice of symmetric families of generators of $\mathcal{H}, \mathcal{G}$, the $\mathcal{H}$-orbits have infinite diameter and the $\mathcal{G}$-orbits have finite diameter, and thus cannot be coarsely quasi-isometric.

In the measure theoretic setting, this problem is solved by considering *Kakutani equiv- alences* [?], which are kind of measure theoretic counterparts of étale equivalences with the additional requirement that the coarsely quasi-isometry type of the orbits is preserved. In the present topological context, the above problem will be addressed without adding more con- ditions to étale equivalences. Instead, appropriate representatives of pseudogroups and sets of generators will be chosen to determine a coarse quasi-isometry type on the orbits. The choice of appropriate pseudogroup representatives is easy, while the choice of appropriate generators is rather delicate.

Let $\mathcal{H}$ be a pseudogroup of local transformations of a locally compact space $Z$ with compact orbit space. By Lemma 2.26, there is a relatively compact open subset $U$ of $Z$ that meets all $\mathcal{H}$ orbits. If $\mathcal{H}$ is indeed compactly generated, the restriction $\mathcal{G}$ of $\mathcal{H}$ to $U$ is a representative of $\mathcal{H}$ whose orbits will be shown to have a canonical coarsely quasi-isometry type, which is determined by any symmetric set $E$ of generators of $\mathcal{G}$ satisfying certain conditions. The first condition on $E$ is that it must be a system of compact generation of $\mathcal{H}$ on $U$. But this is not enough because there are systems of compact generation on the same open set inducing different coarsely quasi-isometry types in the same orbit; such an example will be given in Section 3.6. So a second new condition is introduced as follows.

**Definition 3.22.** A finite symmetric family $E$ of generators of a pseudogroup $\mathcal{H}$ of local transformations of a locally compact space $Z$ is said to be *recurrent* if there exists a relatively compact open subset $U \subset Z$ and some $R > 0$ such that, for any of any $\mathcal{H}$-orbit $\mathcal{O}$, the intersection $\mathcal{O} \cap U$ is an $R$-net in $\mathcal{O}$ with $d_E$; i.e., for any $x \in Z$ there exists some $h \in \mathcal{H}$ so that $x \in \text{dom } h$, $|h|_{E,x} < R$ and $h(x) \in U$.
The role played by $U$ in Definition 3.22 can actually be played by any relatively compact open subset that meets all orbits, as shown by the following result.

**Lemma 3.23.** Let $\mathcal{H}$ be a pseudogroup of local transformations of a locally compact space $Z$, and let $E$ be a recurrent finite symmetric family of generators of $\mathcal{H}$. If $V \subset Z$ is an open set that meets every orbit, then there exists $S > 0$ such that any $d_E$-ball of radius $S$ in any $\mathcal{H}$-orbit meets $V$.

*Proof.* By definition, there exist a relatively compact open subset $U \subset Z$ and a positive number $R$ such that any $d_E$-ball of radius $R$ in any $\mathcal{H}$-orbit meets $U$. Since $V$ also meets every orbit, there exists a finite family $F \subset \mathcal{H}$ such that:

* the sources of elements of $F$ cover the compact closure $\bar{U}$;

* the targets of elements of $F$ are contained in $V$; and

* each element of $F$ is a composite of elements of $E$.

Let $r > 0$ be an integer so that every $g \in F$ can be written as a composition of at most $r$ elements of $E$.

Fix any $x \in Z$. On the one hand, there is some $h \in \mathcal{H}$ with $x \in \text{dom} h$, $|h|_{E,x} < R$ and $h(x) \in U$. On the other hand, there is some $g \in F$ whose domain contains $h(x)$. So $x \in \text{dom}(gh)$, $gh(x) \in V$, and

$$|gh|_{E,x} \le |g|_{E,h(x)} + |h|_{E,x} \le r + R.$$

Thus the result follows with $S = r + R$. $\square$

<figure><img src="image_5.png" /></figure>

Let $\mathcal{H}$ be a compactly generated pseudogroup of local transformations of a locally compact space $Z$. A system of compact generation of $\mathcal{H}$ on a relatively compact subset $U \subset Z$ that meets every orbit is called *recurrent* if it is recurrent when considered as finite symmetric set of generators of the restriction of $\mathcal{H}$ to $U$. An example of a non-recurrent system of compact generation will be given in Section 3.6. On the other hand, the existence of recurrent systems of compact generation will be a consequence of the following result.

**Lemma 3.24.** With the above notation, let $E$ be a system of compact generation of $\mathcal{H}$ on $U$. For each $g \in E$, fix a extension $\bar{g} \in \mathcal{H}$ of $g$ with $\text{dom} \bar{g} \subset \text{dom} \bar{g}$. Suppose that every $x \in \bar{U}$ has an open neighborhood $V_x$ in $Z$ such that

$$V_x \subset \text{dom} \bar{g}_x, \quad V_x \cap U \subset \text{dom} g_x, \quad \overline{g_x(V_x)} \subset U$$

for some $g_x \in E$. Then $E$ is recurrent.

*Proof.* For each $x \in \bar{U}$, let $W_x = \overline{g_x(V_x)}$; its closure can be assumed to be contained in $U$. The compactness of $\bar{U}$ implies that $\bar{U} \subset V_{x_1} \cup \cdots \cup V_{x_n}$, for some finite set of points $x_1, \dots, x_n \in \bar{U}$. Let $V_k = V_{x_k}$, $W_k = W_{x_k}$ and $g_k = g_{x_k}$ for $k = 1, \dots, n$, so $W = W_1 \cup \cdots \cup W_n$ is a relatively compact open subset of $U$. Moreover, each $y \in U$ belongs to some $V_k$; thus $y \in V_k \cap U \subset \text{dom} g_k$ and $g_k(y) \in W$, yielding $d_E(y, W \cap \mathcal{H}(y)) \le 1$. $\square$

<figure><img src="image_6.png" /></figure>
**Corollary 3.25.** Let $\mathcal{H}$ be a compactly generated pseudogroup of local transformations of a locally compact space $Z$, and let $U$ be a relatively compact open subset of $Z$ that meets all $\mathcal{H}$-orbits. Then there exists a recurrent system $E$ of compact generation of $\mathcal{H}$ on $U$ satisfying the following property. The extension $\bar{g} \in \mathcal{H}$ of each $g \in E$ with $\overline{\operatorname{dom} g} \subset \operatorname{dom} \bar{g}$ can be chosen so that $\overline{E} = \{\bar{g} \mid g \in E\}$ is also a recurrent system of compact generation on some relatively compact open subset $U' \subset Z$ with $\overline{U} \subset U'$.

*Proof.* Since $U$ meets every $\mathcal{H}$-orbit and $\overline{U}$ is compact, there exists a finite family $F \subset \mathcal{H}$ satisfying the following properties:

* each $f \in F$ is the restriction of some $\bar{f} \in \mathcal{H}$ whose domain is relatively compact and contains $\overline{\operatorname{dom} f}$;

* each $\bar{f}$ is the restriction of some $\tilde{f} \in \mathcal{H}$ with $\overline{\operatorname{dom} \tilde{f}} \subset \operatorname{dom} \tilde{f}$;

* the sources of elements of $F$ cover $\overline{U}$; and

* $\overline{\operatorname{im} \bar{f}} \subset U$ for every $f \in F$.

For each $f \in F$, let $f'$ denote its restriction

$$U \cap \operatorname{dom} f \to f(U \cap \operatorname{dom} f),$$

let $F' = \{f' \mid f \in F\}$, and set $F'^{-1} = \{f'^{-1} \mid f' \in F'\}$.

For any system $G$ of compact generation of $\mathcal{H}$ on $U$, the set $E = G \cup F' \cup F'^{-1}$ is also a system of compact generation of $\mathcal{H}$ on $U$. Moreover, $E$ satisfies the condition of Lemma 3.24 because

$$\overline{U} \subset \bigcup_{f \in F} \operatorname{dom} f,$$

and $\overline{\operatorname{im} \bar{f}} \subset U$ for every $f \in F$; thus $E$ is recurrent.

Now let

$$U' = \bigcup_{f \in F} \operatorname{dom} \bar{f},$$

which is a relatively compact open subset of $Z$ containing $\overline{U}$. Let $\overline{F} = \{\bar{f} \mid f \in F\}$ and $\overline{F}^{-1} = \{\bar{f}^{-1} \mid f \in F\}$, which are subsets of the restriction $\mathcal{G}'$ of $\mathcal{H}$ to $U'$. The extensions $\bar{g}$ of the maps $g \in G$ satisfying $\overline{\operatorname{dom} g} \subset \operatorname{dom} \bar{g}$ can obviously be chosen so that:

* each $\bar{g}$ has source and range in $U'$;

* the set $\overline{\mathcal{G}} = \{\bar{g} \mid g \in G\}$ is symmetric; and

* each $\bar{g}$ is the restriction of some $\tilde{g} \in \mathcal{H}$ with $\overline{\operatorname{dom} \tilde{g}} \subset \operatorname{dom} \tilde{g}$.

Then $\overline{E} = \overline{\mathcal{G}} \cup \overline{F} \cup \overline{F}^{-1}$ is a finite symmetric subset of $\mathcal{G}'$ which generates $\mathcal{G}'$ because $G$ generates $\mathcal{G}$ and $\overline{\operatorname{im} \bar{f}} \subset U$ for all $f \in F$. The above properties guarantee that $\overline{E}$ is a system of compact generation of $\mathcal{H}$ on $U'$. Finally, $\overline{E}$ is recurrent by Lemma 3.24 since $\overline{\operatorname{im} \bar{f}} \subset U$ for all $f \in F$. $\square$
The following is the promised result that shows the invariance of the coarsely quasi-isometry type of the orbits by equivalences when appropriate representatives of pseudogroups and generators are chosen.

**Theorem 3.26.** Let $\mathcal{H}, \mathcal{H}'$ be compactly generated pseudogroups of local transformations of locally compact spaces $Z, Z'$, and let $U, U'$ be relatively compact open subsets of $Z, Z'$ that meet all orbits of $\mathcal{H}, \mathcal{H}'$, respectively. Let $\mathcal{G}, \mathcal{G}'$ denote the restrictions of $\mathcal{H}, \mathcal{H}'$ to $U, U'$, and let $E, E'$ be recurrent symmetric systems of compact generation of $\mathcal{H}, \mathcal{H}'$ on $U, U'$, respectively. Suppose that there exists an equivalence $\mathcal{H} \to \mathcal{H}'$, and consider the induced equivalence $\mathcal{G} \to \mathcal{G}'$ and homeomorphism $U/\mathcal{G} \to U'/\mathcal{G}'$. Then the $\mathcal{G}$-orbits with $d_E$ are equi-coarsely quasi-isometric to the corresponding $\mathcal{G}'$-orbits with $d_{E'}$.

In other words, Theorem 3.26 asserts that, for pseudogroups $\mathcal{H}$ of local transformations of locally compact spaces $Z$ with given sets of generators $E$, the coarsely quasi-isometry type of the orbits is uniformly invariant by equivalences when the following conditions are satisfied. First, $\mathcal{H}$ must be the restriction of a pseudogroup $\mathcal{H}'$ acting on a larger locally compact space where $Z$ is open, relatively compact and meets all $\mathcal{H}'$-orbits. Second, $E$ must be a recurrent system of compact generation of $\mathcal{H}'$ on $Z$.

In order to prove Theorem 3.26, the following preliminary results will be required.

**Lemma 3.27.** Let $\mathcal{H}$ be a compactly generated pseudogroup of local transformations of a locally compact space $Z$, let $U, U'$ be relatively compact open subsets of $Z$ such that $U \cap U'$ meets all $\mathcal{H}$-orbits, and let $E, E'$ be recurrent systems of compact generation of $\mathcal{H}$ over $U, U'$, respectively. Then, for any open set $V$ that meets all $\mathcal{H}$-orbits and with $\bar{V} \subset U \cap U'$, there exists some $C \ge 1$ such that

$$ \frac{1}{C} d_{E'}(x, y) \le d_E(x, y) \le C d_{E'}(x, y) $$

for all $x, y \in V$ lying in the same $\mathcal{H}$-orbit.

*Proof.* Let $\mathcal{G}$ denote the restriction of $\mathcal{H}$ to $U$. By Lemma 3.23, there exists some integer $R > 0$ such that $V$ meets all $d_E$-balls of radius $R$ in all $\mathcal{G}$-orbits. Let $\Phi \subset \mathcal{G}$ denote the finite set of restrictions of the form

$$ g : V \cap \mathrm{dom} g \to g(V \cap \mathrm{dom} g), $$

where $g$ runs over the composites of at most $R$ elements of $E$. It is noted that the images of elements of $\Phi$ cover $U$. Moreover, it may be assumed that $R \ge 2$, and thus that the identity map of $V \cap \mathrm{dom} g$ belongs to $\Phi$ for all $g \in E$ with $V \cap \mathrm{dom} g \ne \emptyset$.

Let $F$ denote the finite set of composites $\psi^{-1} \circ g \circ \phi$, where $\phi, \psi \in \Phi$ and $g \in E$. Observe that each $f \in F$ is the restriction of some $\bar{f} \in \mathcal{G}$ with $\overline{\mathrm{dom} \bar{f}} \subset \mathrm{dom} \bar{f}$. Furthermore, for each $x \in \mathrm{dom} \bar{f}$, it holds that $|\bar{f}|_{E',y} \le |\bar{f}|_{E',x}$, for all $y$ in some neighborhood of $x$ in $\mathrm{dom} \bar{f}$. Hence, since $F$ is finite and the domain of each $f \in F$ is relatively compact in $U'$, there exists an integer $S > 0$ such that $|f|_{E',x} \le S$ for all $f \in F$ and all $x \in \mathrm{dom} f$.
Let $x, y \in V$ be points in the same $\mathcal{H}$-orbit. If $x = y$, then $d_E(x, y) = d_{E'}(x, y) = 0$, and the statement holds trivially with any $C \ge 1$. If $x \ne y$, then $d_E(x, y) = k \ge 1$. Let $g \in \mathcal{G}$ be such that $x \in \text{dom } g$, $g(x) = y$ and $|g|_{E,x} = k$. This element $g$ may be assumed to be of the form $g = g_k \circ \cdots \circ g_1$ for some $g_1, \dots, g_k \in E$. Then, for $i = 1, \dots, k-1$, there exists $\phi_i \in \Phi$ whose image contains $x_i = g_i \circ \cdots \circ g_1(x)$ and such that $z_i = \phi_i^{-1}(x_i) \in V$. Such a map $g$ is equal to a composite $f_k \circ \cdots \circ f_1$ around $x$, where $f_1, \dots, f_k$ are the elements of $F$ given by

$$f_1 = \phi_1^{-1} \circ g_1, \quad f_k = g_k \circ \phi_{k-1}, \quad f_i = \phi_i^{-1} \circ g_i \circ \phi_{i-1},$$

for $i = 2, \dots, k-1$. Observe that $x \in \text{dom } f_1$ and $z_i \in \text{dom } f_{i+1}$ for all $i = 1, \dots, k-1$. Therefore,

$$|g|_{E',x} \le |f_k|_{E',z_{k-1}} + \dots + |f_2|_{E',z_1} + |f_1|_{E',x} \le kS,$$

yielding

$$d_{E'}(x, y) \le S d_E(x, y).$$

Similarly,

$$d_E(x, y) \le S' d_{E'}(x, y)$$

for some integer $S' > 0$, and result follows with $C = \max\{S, S'\}$. $\square$

**Corollary 3.28.** Let $\mathcal{H}$ be a compactly generated pseudogroup of local transformations of a locally compact space $Z$ and let $U, U'$ be relatively compact open subsets of $Z$ such that $U \cap U'$ meets all $\mathcal{H}$-orbits. Let $\mathcal{G}, \mathcal{G}'$ denote the restrictions of $\mathcal{H}$ to $U, U'$, and let $E, E'$ be recurrent systems of compact generation of $\mathcal{H}$ over $U, U'$, respectively. Then the $\mathcal{G}$-orbits with $d_E$ are equi-coarsely quasi-isometric to the corresponding $\mathcal{G}$'-orbits with $d_{E'}$.

*Proof.* There exists an open set $V$ meeting every $\mathcal{H}$-orbit and such that $\bar{V} \subset U \cap U'$. By Lemma 3.23, there also exist $R, R' > 0$ such that any $d_E$-ball of radius $R$ in any $\mathcal{G}$-orbit meets $V$, and any $d_{E'}$-ball of radius $R'$ in any $\mathcal{G}$'-orbit meets $V$. So the result follows from Lemma 3.27. $\square$

*Proof of Theorem 3.26.* Let $\Phi: \mathcal{H} \to \mathcal{H}'$ be an equivalence, and let $\mathcal{H}''$ be the pseudogroup of local transformations of $Z'' = Z \sqcup Z'$ generated by $\mathcal{H} \cup \mathcal{H}' \cup \Phi$, considered as sets of local transformations of $Z''$ in the obvious way. Then $U, U'$ and $U'' = U \sqcup U'$ are relatively compact open subsets of $Z''$ that meet all $\mathcal{H}''$-orbits. The restrictions of $\mathcal{H}''$ to $U$ and $U'$ are $\mathcal{G}$ and $\mathcal{G}'$, thus $E$ and $E'$ are recurrent systems of compact generation of $\mathcal{H}''$ on $U$ and $U'$, respectively. Let $\mathcal{G}''$ denote the restriction of $\mathcal{H}''$ to $U''$, and select a recurrent system $E''$ of compact generation of $\mathcal{H}''$ on $U''$. By Corollary 3.28, the $\mathcal{G}''$-orbits with $d_{E''}$ are equi-coarsely quasi-isometric to the corresponding $\mathcal{G}$-orbits with $d_E$, and also to the corresponding $\mathcal{G}'$-orbits with $d_{E'}$. $\square$

The following is a slight sharpening of Lemma 3.27 that will be used later.
**Proposition 3.29.** Let $\mathcal{H}$ be a compactly generated pseudogroup of local transformations of a locally compact space $Z$, let $U, U'$ be relatively compact open subsets of $Z$ such that $U \cap U'$ meets all $\mathcal{H}$-orbits, and let $E, E'$ be recurrent systems of compact generation of $\mathcal{H}$ over $U, U'$, respectively. Then there exists some $C \ge 1$ such that

$$ \frac{1}{C} d_{E'}(x, y) \le d_{E}(x, y) \le C d_{E'}(x, y) $$

for all $x, y \in U \cap U'$ lying in the same $\mathcal{H}$-orbit.

*Proof.* Assume first that $U \subset U'$ and $E \subset E'$. Hence the first inequality of the statement holds for any $C \ge 1$.

Let $\mathcal{G}$ denote the restriction of $\mathcal{H}$ to $U$. Fix any open set $V$ that meets all $\mathcal{H}$-orbits and with $\bar{V} \subset U$. Then, by Lemma 3.23, there is some $R > 0$ such that $\mathcal{O} \cap V$ is an $R$-net in $(\mathcal{O}, d_E)$ for every $\mathcal{G}$-orbit $\mathcal{O}$.

Take arbitrary points $x, y \in U$ lying in the same $\mathcal{H}$-orbit. We can assume that $x \ne y$, otherwise the inequalities of the statement hold trivially for any $C \ge 1$. There are points $x', y' \in V$ with $d_E(x, x'), d_E(y, y') \le R$. So $d_{E'}(x, x'), d_{E'}(y, y') \le R$ as well because $E \subset E'$. By Lemma 3.27, there is some $K \ge 1$, independent of $x', y'$, such that

$$ \frac{1}{K} d_{E'}(x', y') \le d_E(x', y') \le K d_{E'}(x', y') . $$

Therefore

$$ \begin{aligned} d_E(x, y) &\le d_E(x, x') + d_E(x', y') + d_E(y, y') \\ &\le d_E(x', y') + 2R \\ &\le K d_{E'}(x', y') + 2R \\ &\le K (d_{E'}(x', x) + d_{E'}(x, y) + d_{E'}(y, y')) + 2R \\ &\le K (d_{E'}(x, y) + 2R) + 2R \\ &\le (K + 2KR + 2R) d_{E'}(x, y) \end{aligned} $$

since $d_{E'}(x, y) \ge 1$, yielding the second inequality of the statement with $C = K + 2KR + 2R$.

In the general case, let $E_0$ be a recurrent system of compact generation of $\mathcal{H}$ over $U \cap U'$. Then $F = E_0 \cup E$ and $F' = E_0 \cup E'$ are recurrent systems of compact generation of $\mathcal{H}$ over $U$ and $U'$, respectively. It has been proved that there are some constants $K, K', L, L' \ge 1$ such that

$$ \begin{aligned} \frac{1}{K} d_F(x, y) &\le d_{E_0}(x, y) \le K d_F(x, y), & \frac{1}{K'} d_{F'}(x, y) &\le d_{E_0}(x, y) \le K' d_{F'}(x, y), \\ \frac{1}{L} d_F(x, y) &\le d_E(x, y) \le L d_F(x, y), & \frac{1}{L'} d_{F'}(x, y) &\le d_{E'}(x, y) \le L' d_{F'}(x, y) \end{aligned} $$

for all $x, y \in U \cap U'$ lying in the same $\mathcal{H}$-orbit. Thus the result follows with $C = KK'LL'$. $\square$
Let $\mathcal{H}$ be a compactly generated pseudogroup of local transformations of a locally compact space $Z$, let $U$ be a relatively compact open subset of $Z$ that meets all $\mathcal{H}$-orbits, and let $\mathcal{G}$ denote the restriction of $\mathcal{H}$ to $U$. By considering the identity map on the $\mathcal{G}$-orbits and inclusions of systems of compact generation on $U$, we get an inductive system of metric spaces. Note also that distances between points in the same $\mathcal{G}$-orbit do not increase by considering larger systems of compact generation. By Theorem 3.26, the corresponding “inductive system of coarse quasi-isometry types” has a limit, which is uniformly reached just when a recurrent system of compact generation is considered. In fact, the following consequence of Proposition 3.29 shows that it happens so with the corresponding “inductive system of lipschitz types.”

**Corollary 3.30.** With the above notation, let $E$ be a recurrent symmetric system of compact generation of $\mathcal{H}$ on $U$. Any other symmetric system $E'$ of compact generation of $\mathcal{H}$ on $U$ is recurrent if and only if there exists some $C \ge 1$ such that

$$ \frac{1}{C} d_{E'}(x, y) \le d_{E}(x, y) \le C d_{E'}(x, y) $$

for all $x, y \in U$ lying in the same $\mathcal{G}$-orbit.

*Proof.* The “only if” part holds by Proposition 3.29.

Now assume that the inequalities of the statement hold for some $C \ge 1$ and all $x, y \in U$ lying in the same $\mathcal{G}$-orbit. Take any open set $V$ that meets all $\mathcal{H}$-orbits and with $\bar{V} \subset U$. By Lemma 3.23, there is some $R > 0$ such that $\emptyset \cap V$ is an $R$-net in $(\emptyset, d_E)$ for every $\mathcal{G}$-orbit $\emptyset$. It easily follows that $\emptyset \cap V$ is an $CR$-net in $(\emptyset, d_{E'})$ for every $\mathcal{G}$-orbit $\emptyset$, and thus $E'$ is recurrent. $\square$
## 3.5 Fölner orbits and invariant measures

The Fölner condition will be used in the next section to distinguish coarse quasi-isometry types of orbits. The property that fölner orbits give rise to invariant measures will be also needed. This was shown by S. Goodman and J. Plante [?] for pseudogroups acting on compact metric spaces, and is partially improved in this section by using recurrent compact generation instead of a compact space. For compact foliated spaces, it is well known that fölner leaves induce invariant transverse probability measures. So recurrence can be useful to show that compactly generated pseudogroups behave like compact foliated spaces, which is in the spirit of a famous project of A. Haefliger [?].

Let $M$ be a metric space with metric $d$. A quasi-lattice $\Gamma$ of $M$ is a net of $M$ such that for every $r \ge 0$ there is some $K_r \ge 0$ such that $\#(\Gamma \cap B(x,r)) \le K_r$ for every $x \in M$. Not every metric space has a quasi-lattice, but metric spaces with bounded complexity in a reasonable sense do; see e.g. [?] for examples. The metric space $M$ is said to be of coarse bounded geometry if it has a quasi-lattice.

For any $r > 0$, the $r$-boundary of each subset $S \subset M$ is the set

$$ \partial_r S = \{x \in S \mid d(x, S) < r \text{ and } d(x, M \setminus S) < r\}. $$

The notation $\partial_r^M S$ will be also used for $\partial_r S$. Then $M$ is called amenable [?] if it has a quasi-lattice $\Gamma$ and a sequence of finite subsets $S_n \subset \Gamma$ such that

$$ (3.8) \qquad \lim_{n \to \infty} \frac{\# \partial_r^\Gamma S_n}{\# S_n} = 0 $$

for each $r > 0$. Such a sequence $S_n$ will be called a fölner sequence in $\Gamma$. Since

$$ \partial_r^\Gamma S \setminus S \subset \bigcup_{x \in S \cap \partial_r^\Gamma S} (\Gamma \cap B(x, r)) $$

for every $S \subset \Gamma$, it follows that

$$ (3.9) \qquad \#(\partial_r S \setminus S) \le K_r \cdot \#(S \cap \partial_r S) $$

if $\#(\Gamma \cap B(x, r)) \le K_r$ for any $x \in \Gamma$. Therefore the amenability condition (3.8) is equivalent to

$$ \lim_{n \to \infty} \frac{\#(S_n \cap \partial_r^\Gamma S_n)}{\# S_n} = 0 $$

for each $r > 0$.

**Theorem 3.31** (Block-Weinberger [?]). Let $M, M'$ be roughly equivalent metric spaces of coarse bounded geometry. Then $M$ is amenable if and only if so is $M'$.

This result was proved in [?] in the following way. First, the uniformly finite homology $H_\bullet^{\text{uf}}(M)$ is introduced for any metric space $M$. Second, it is shown that, if two metric spaces $M, M'$ are roughly equivalent then $H_\bullet^{\text{uf}}(M) \cong H_\bullet^{\text{uf}}(M')$. Finally, it is shown that a
metric space $M$ of coarse bounded geometry is amenable if and only if $H_0^{\text{uf}}(M) \neq 0$, and
Theorem 3.31 follows. In particular, amenability is a coarse quasi-isometry invariant for
metric spaces of coarse bounded geometry, which also has the following direct proof. ***

The following lemma will be useful in the proof of the main result of this section.

**Lemma 3.32.** Let $\Gamma$ be a quasi-lattice in some metric space, and let $A$ be a $C$-net in $\Gamma$ for some $C > 1$. Fix any $K > 0$ such that every ball of radius $C$ in $\Gamma$ has at most $K$ points. Then

$$ \#S \le \#(S \cap \partial_C^\Gamma S) + K \cdot \#(A \cap S) $$

for any $S \subset \Gamma$.

*Proof.* For every $x \in S$, there is some $a \in A$ so that $d(x, a) < C$, and thus $x \in S \cap \partial_C^\Gamma S$ if $a \notin S$. Therefore

$$ S \subset (S \cap \partial_C^\Gamma S) \cup \bigcup_{a \in A \cap S} (\Gamma \cap B(a, C)), $$

yielding the inequality of the statement. $\square$

We are interested in the case of a metric space $M$ whose points are the vertices of some
connected graph, and where the distance between two points is the minimum number of
contiguous edges needed to join them. In this case, besides the amenability condition of [?],
$M$ may be also fölner in the usual graph sense, which is defined as follows. The boundary $\partial S$
of any $S \subset M$ is the set of points $x \in S$ such that there is some edge joining $x$ with some
point in $M \setminus S$; i.e., $\partial S = S \cap \partial_2 S$. Then $M$ is fölner (as a graph) when there is a sequence
of finite subsets $S_n \subset M$ such that

$$ \lim_{n \to \infty} \frac{\#\partial S_n}{\#S_n} = 0. $$

Note that $M$ is a quasi-lattice in itself just when it is a graph of finite type; i.e., there
is a uniform upper bound $K$ on the number of edges that meet at every vertex; indeed,
$\#B(x, r) \le K^r$ for all $r \ge 0$ when there is such a $K$. In this case, since

$$ \partial_r S \subset \bigcup_{x \in \partial S} B(x, r), $$

it follows that

$$ (3.10) \qquad \#\partial_r S \le K^r \cdot \#\partial S $$

for any $r > 0$. Hence a graph of finite type is amenable (as metric space) if and only if it is
fölner (in the graph sense).

Consider again a pseudogroup $\mathcal{H}$ of local transformations of a space $Z$ with the metric
$d_E$ on the orbits induced by a finite symmetric set $E$ of generators of $\mathcal{H}$. Then we get a
graph by introducing an edge between two points $x, y \in Z$ whenever there is some $g \in E$
with $g(x) = y$. Thus $\#E$ is an upper bound for the number of edges that meet at every
vertex. Observe that each orbit of $\mathcal{H}$ is given by the vertices of a connected component of this graph, and $d_E$ is the metric induced by this graph on its connected components. The following notation and terminology will be used in this setting:

* Let $\partial^E S$ denote the boundary of any finite subset $S$ of an orbit with respect to the graph structure induced by $E$;

* for $r > 0$, let $\partial_r^E S$ denote the $r$-boundary of any finite subset $S$ of an orbit with respect to the metric $d_E$;

* a fölner sequence of an orbit with the metric $d_E$ (or with the graph structure induced by $E$) will be called an E-fölner sequence; and

* an orbit with an E-fölner sequence will be called E-fölner or E-amenable.

**Theorem 3.33.** Let $\mathcal{H}$ be a compactly generated pseudogroup of local transformations of a locally compact metric space $Z$, let $U$ be a relatively compact open subset of $Z$ that meets all $\mathcal{H}$-orbits, and let $\mathcal{G}$ be the restriction of $\mathcal{H}$ to $U$. Consider the metric on the $\mathcal{G}$-orbits induced by a recurrent symmetric system $E$ of compact generation of $\mathcal{H}$ on $U$. If some $\mathcal{G}$-orbit is E-fölner, then there is a non-trivial non-negative $\mathcal{G}$-invariant borelian measure on $U$ of finite mass.

*Proof.* Let $S_n$ be an E-fölner sequence of some orbit $\mathcal{O}$ of $\mathcal{G}$. As in [?], a measure $\mu$ is constructed on $U$ as a limit of averaging measures on the finite sets $S_n$. Let $C_0(U)$ be the banach space of continuous functions $f: U \to \mathbb{R}$ that vanish at infinity, endowed with the supremum norm $\|\|$ given by

$$ \|f\| = \sup_{x \in U} |f(x)| $$

For each $n \in \mathbb{N}$, let $\mu_n: C_0(U) \to \mathbb{R}$ be defined by

$$ \mu_n(f) = \frac{1}{\#S_n} \sum_{x \in S_n} f(x) $$

for $f \in C_0(U)$. Each $\mu_n$ is obviously linear and continuous; i.e., it is an element of the (algebraic-topological) dual space $C_0(U)'$. Moreover it is easy to check that

$$ |\mu_n(f)| \le \|f\| $$

for all $n \in \mathbb{N}$ and $f \in C_0(U)$. Therefore, by the Banach-Alaoglu theorem, the set $\{\mu_n | n \in \mathbb{N}\}$ is relatively compact in $C_0(U)'$ with the weak* topology; i.e., the topology of pointwise convergence. Then, by passing to a subsequence if necessary, we can assume that the sequence $\mu_n$ is pointwisely convergent to some $\mu$ in $C_0(U)'$, which can be considered as a borelian measure of finite mass on $U$ by the representation theorem of Riesz. This $\mu$ is non-negative since all the $\mu_n$ are probability measures. The $\mathcal{G}$-invariance of $\mu$ follows from the E-fölner condition of the sequence $S_n$ since, as shown in [?],

$$ |\mu(f \circ g) - \mu(f)| \le 2 \|f\| \lim_{n} \frac{\#\partial^E S_n}{\#S_n} $$
for all $g \in E$ and $f \in C_0(U)$ with $\text{supp} f \subset \text{im} g$.

Finally, we show that $\mu$ is not trivial. Take any open set $U'$ that meets all $\mathcal{G}$-orbits and with $\overline{U'} \subset U$, and consider any non-negative function $f \in C_0(U)$ with $f(x) = 1$ for all $x \in U'$. By Lemma 3.23, there is some $C > 0$ such that $\mathcal{O} \cap U'$ is a $C$-net in $\mathcal{O}$ with $d_E$. Then

$$ \mu(f) \geq \lim_{n} \frac{\#(S_n \cap U')}{\#S_n} \geq (\#E)^{-C} \lim_{n} \frac{\#S_n - \#(S_n \cap \partial_C^E S_n)}{\#S_n} = (\#E)^{-C} > 0 $$

by Lemma 3.32 and since the sequence $S_n$ is $E$-fölner. $\square$

**Remarks.** (i) In the proof of Theorem 3.33, the measure $\mu$ could be trivial if $E$ were not recurrent. This is different from the arguments of [?] because $U$ is not compact. For instance, for the pseudogroup on $\mathbb{R}$ generated by the translation $g(x) = x+1$, the sets $\{n, n+1, \dots, 2n\}$ ($n \in \mathbb{N}$) form a $\{g, g^{-1}\}$-fölner sequence in an orbit, and the limit of corresponding averaging measures is trivial.

(ii) The statement of Theorem 3.33 could be more general. As in [?], the definitions and arguments could be modified to remove the condition that all sets of the fölner sequence lie in the same orbit: it would be enough to have what is called an *averaging sequence* in [?]. But our result is simpler and general enough for our purposes in the next section.

## 3.6 An example of non-recurrent compact generation

The purpose of this section is to give an example showing that there exist non-recurrent systems of compact generation, and that the coarsely quasi-isometry type of the orbits may depend on the system of compact generation if recurrence is not considered.

Fix real numbers

$$ a < a' < a'' < b'' < b' < b, $$

and choose homeomorphisms $\phi : \mathbb{R} \to (a, b)$ and $\tilde{g}_1 : \mathbb{R} \to \mathbb{R}$ satisfying the following properties:

* $\phi(x) = x$ for all $x \in [a'', b'']$;

* $\phi(x) > x$ for all $x \in (-\infty, a'')$;

* $\phi(x) < x$ for all $x \in (b'', \infty)$;

* $\phi(a) = a'$ and $\phi(b) = b'$;

* $\tilde{g}_1(x) = x$ for all $x \in (-\infty, a] \cup [b, \infty)$;

* $\tilde{g}_1(x) > x$ for all $x \in (a, b)$; and

* $\tilde{g}_1(a'') < b''$.
Let $\mathcal{H}$ be the pseudogroup of local transformations of $\mathbb{R}$ generated by $\phi$ and $\tilde{g}_1$. The bounded open interval $U = (a, b)$ meets all $\mathcal{H}$-orbits because $\phi(\mathbb{R}) = U$, and let $\mathcal{G}$ denote the restriction of $\mathcal{H}$ to $U$.

Now define a map $\tilde{g}_2 : \mathbb{R} \to \mathbb{R}$ by setting

$$ \tilde{g}_2(x) = \begin{cases} \phi \circ \tilde{g}_1 \circ \phi^{-1}(x) & \text{if } a < x < b, \\ x & \text{otherwise.} \end{cases} $$

Such a $\tilde{g}_2$ is a homeomorphism and satisfies the following properties:

* $\tilde{g}_2(x) = x$ for all $x \in (-\infty, a'] \cup [b', \infty)$;

* $\tilde{g}_2(x) > x$ for all $x \in (a', b')$; and

* $\phi \circ \tilde{g}_1(x) = \tilde{g}_2 \circ \phi(x)$ for all $x \in U$.

We now prove that $\mathcal{G}$ is generated by the restrictions $\tilde{g}_1, \tilde{g}_2 : U \to U$, which will be denoted by $g_1, g_2$. It is enough to prove that the restriction $\phi : U \to \phi(U)$ is in the pseudogroup $\mathcal{G}'$ generated by $g_1, g_2$. Since $\tilde{g}_1(a'') < b'',$ the collection of sets $U_n = g_1^n(a'', b'')$, $n \in \mathbb{Z}$, is an open covering of $U$, and thus it suffices to prove that each restriction $\phi : U_n \to \phi(U_n)$ is in $\mathcal{G}'$. But $\phi$ is the identity on $(a'', b'') = g_1^{-n}(U_n)$, yielding

$$ g_1^{-n} = \phi \circ g_1^{-n} = g_2^{-n} \circ \phi $$

on $U_n$. So $\phi = g_2^n \circ g_1^{-n}$ on $U_n$, which belongs to $\mathcal{G}'$, as desired.

Since $\text{dom } g_i \subset \text{dom } \tilde{g}_i$ for $i = 1, 2$, it follows that $E = \{g_1, g_2, g_1^{-1}, g_2^{-1}\}$ is a system of compact generation of $\mathcal{H}$ on $U$. For the open subset $V = (a', b) \subset U$, it will be shown that

$$ (3.11) \qquad \lim_{x \to a} d_E(x, V \cap \mathcal{G}(x)) \to \infty. $$

Since $V$ meets every $\mathcal{G}$-orbit, it follows from Lemma 3.23 and (3.11) that $E$ is not recurrent. To prove (3.11), let

$$ \nu(x) = \min\{n \in \mathbb{N} \mid g_1^n(x) \in V\} $$

for each $x \in U$. Clearly,

$$ x \in V \iff \nu(x) = 0, \quad x < y \implies \nu(x) \ge \nu(y), \quad \lim_{x \to a} \nu(x) = \infty. $$

Take any $x \in U$ and some $h \in \mathcal{G}$ with

$$ x \in \text{dom } h, \quad h(x) \in V, \quad |h|_{E,x} = d_E(x, V \cap \mathcal{G}(x)). $$

For $n = |h|_{E,x}$, we have $h = g_{i_n}^{\epsilon_n} \circ \dots \circ g_{i_1}^{\epsilon_1}$ around $x$ for some $i_1, \dots, i_n \in \{1, 2\}$ and $\epsilon_1, \dots, \epsilon_n \in \{\pm 1\}$. Let $x_k = g_{i_k}^{\epsilon_k} \circ \dots \circ g_{i_1}^{\epsilon_1}(x)$ for every $k = 1, \dots, n$. Since $x_k \notin V$ for each $k < n$, either $x_{k+1} \le x_k$ (yielding $\nu(x_{k+1}) \ge \nu(x_k)$), or $g_{i_{k+1}}^{\epsilon_{k+1}} = g_1$ (yielding $\nu(x_{k+1}) = \nu(x_k) - 1$). Therefore

$$ \nu(x) \le n = d_E(x, V \cap \mathcal{G}(x)) $$
because $\nu(x_n) = 0$, and (3.11) follows.

Finally, let $F$ be a recurrent symmetric system of compact generation of $\mathcal{H}$ on $U$. We will show that no $\mathcal{G}$-orbit with the metric $d_E$ is coarsely quasi-isometric to itself with the metric $d_F$. Suppose that this is not true for some $\mathcal{G}$-orbit $\mathcal{O}$. Since the open interval $I = (a, a')$ meets every orbit and since $g_2$ is the identity on $I$, there is some point $x_0 \in \mathcal{O} \cap I$ such that the set

$$X = \{g_1^{-m}(x_0) | m \in \mathbb{N}\} \subset \mathcal{O} \cap I$$

satisfies $\partial^E X = \{x_0\}$. Hence $\mathcal{O}$ is $E$-fölner; for instance, an $E$-fölner sequence for $\mathcal{O}$ is given by the sets

$$S_n = \{g_1^{-m}(x_0) | 0 \le m \le n\}.$$

Then $\mathcal{O}$ is $F$-fölner too by Theorem 3.31 since we are assuming that the metrics $d_E, d_F$ on $\mathcal{O}$ are coarsely quasi-isometric. So, by Theorem 3.33 and because $F$ is recurrent, there is a non-trivial non-negative $\mathcal{G}$-invariant borelian measure $\mu$ on $U$ of finite mass. Fix any $t \in U$, and let $I_n = (a, g_1^n(t))$ for each $n \in \mathbb{Z}$. Since $\mu$ is $\mathcal{G}$-invariant and $g(I_n) = I_{n+1}$, all sets $I_n$ have the same $\mu$-measure, which is finite since $\mu$ has finite mass. Then $\mu(I_{n+1} \setminus I_n) = 0$ for all $n$, whence $\mu(U) = 0$ because $U = \bigcup_n(I_{n+1} \setminus I_n)$. This is a contradiction because $\mu$ is non-trivial and non-negative.

## 3.7 Good metrics on the leaves

Let $(X, \mathcal{F})$ be a compact foliated space. Compact generation and recurrence are properties satisfied by its holonomy pseudogroup with the generators given by an apropriate finite defining cocycle. To see this, fix a finite regular *** atlas $\mathcal{U} = \{U_i, \phi_i\}$ with $\phi_i : U_i \to D_i \times Z_i$. Suppose also that $\mathcal{U}$ is a refinement of another finite regular atlas $\tilde{\mathcal{U}}$. Indeed, we can assume that $\mathcal{U}$ is a shrinking of $\tilde{\mathcal{U}}$; i.e., $\tilde{\mathcal{U}} = \{\tilde{U}_i, \tilde{\phi}_i\}$ with $\tilde{\phi}_i : \tilde{U}_i \to \tilde{Z}_i \times \tilde{D}_i$ so that $\overline{U_i} \subset \tilde{U}_i, \overline{Z_i} \subset \tilde{Z}_i, \overline{D_i} \subset \tilde{D}_i$, and $\phi_i$ is the restriction of $\tilde{\phi}_i$.

Let $\{U_i, p_i, h_{i,j}\}$ and $\{\tilde{U}_i, \tilde{p}_i, \tilde{h}_{i,j}\}$ be the corresponding defining cocycles of $\mathcal{F}$, and let $\mathcal{H}$ and $\tilde{\mathcal{H}}$ be the corresponding representatives of the holonomy pseudogroup of $\mathcal{F}$. Thus $p_i : U_i \to Z_i$ is a restriction of $\tilde{p}_i : \tilde{U}_i \to \tilde{Z}_i, h_{i,j} : Z_{i,j} \to Z_{j,i}$ is a restriction of $\tilde{h}_{i,j} : \tilde{Z}_{i,j} \to \tilde{Z}_{j,i}$, $\mathcal{H}$ is generated by the local transformations $h_{i,j}$ of $Z = \bigsqcup_i Z_i$, and $\tilde{\mathcal{H}}$ is generated by the local transformations $\tilde{h}_{i,j}$ of $\tilde{Z} = \bigsqcup_i \tilde{Z}_i$. It follows that $Z$ has compact closure in $\tilde{Z}$, $\mathcal{H}$ is the restriction of $\tilde{\mathcal{H}}$ to $Z$, $Z$ meets all $\tilde{\mathcal{H}}$-orbits, and $E = \{h_{i,j}\}$ is a system of compact generation of $\tilde{\mathcal{H}}$ on $Z$.

**Proposition 3.34.** *E is recurrent.*

*Proof.* Fix any point $x$ in the closure of some $Z_i$ in $\tilde{Z}_i$, and take some $U_j$ meeting $\tilde{p}_i^{-1}(x)$. Then $V_{i,j} = \tilde{p}_i(\tilde{U}_i \cap U_j)$ is a neighborhood of $x$ in $\tilde{Z}_i$ such that $V_{i,j} \subset \tilde{Z}_{i,j}$, $V_{i,j} \cap Z_i = Z_{i,j}$ and $\tilde{h}_{i,j}(V_{i,j}) \subset Z_j$. Hence the recurrency of $E$ follows from Lemma 3.24. $\square$
Therefore, according to Theorem 3.26, the coarsely quasi-isometry type of the $\mathcal{H}$-orbits with the metric $d_E: R_{\mathcal{H}} \to [0, \infty)$ induced by $E$ is uniformly independent of the defining cocycle $\{U_i, p_i, h_{i,j}\}$ satisfying the above conditions.

Recall that each $\tilde{Z}_i$ can be considered as a local transversal of $\mathcal{F}$ via the map

$$ \tilde{Z}_i = \{y_i\} \times \tilde{Z}_i \subset \tilde{D}_i \times \tilde{Z}_i \xrightarrow{\tilde{\phi}_i^{-1}} \tilde{U}_i \subset X, $$

where $y_i$ is any fixed point in $D_i$, and all of these local transversals can be assumed to be disjoint from each other, yielding an embedding of $\tilde{Z}$ into $X$ as a complete transversal. Since each $\mathcal{H}$-orbit injects into the corresponding leaf in this way, the above coarsely quasi-isometry types of the $\mathcal{H}$-orbits can be somehow considered as coarsely quasi-isometry types of the leaves. This can be made more explicit as follows.

If $\mathcal{F}$ is of class $C^1$, then there exists a metric tensor on the leaves that is continuous on $X$. In this case, it is well known that the above injection of $\mathcal{H}$-orbits into corresponding leaves is a collection of equi-coarse quasi-isometries. This is just a consequence of having a uniform upper bound of the diameter of the plaques for the given finite defining cocycle. See e.g. [?] for a more detailed description of the coarse quasi-isometry between the leaves and the orbits of the holonomy pseudogroup, which is already implicit in Plante [?]. The bornotopy type of leaves is studied in Hurder [?], and most of its discussion also applies to the coarse quasi-isometry type.

Even without any smoothness condition, there is a class of metrics on $R_{\mathcal{F}}$ so that the leaves are uniformly coarsely quasi-isometric to the corresponding orbits of $\mathcal{H}$. First, a uniform structure on $X$ can be defined by requiring that the plaques of any regular *** finite atlas of the foliated space $X$ form a uniform cover; thus it induces the leaf topology of $X$. This will be called the *leaf uniformity* of $X$. Notice that this uniform structure induces the topology of the leaves.

**Proposition 3.35.** *X is a complete uniform space with the leaf uniformity.*

*Proof.* By definition of the leaf uniformity, for any finite regular atlas of $X$, each Cauchy sequence $x_n$ in $X$ is contained in a finite union of plaques, which has compact closure in $X$ with the leaf topology. So $x_n$ is convergent in $X$ with the leaf topology. $\square$

The foliated structure also induces an equi-coarse quasi-isometry type *** on its leaves as follows. The following notation will be used. For subsets $A, B \subset X \times X$, the composite $B \circ A$ is the set of points $(x, z) \in X \times X$ such that there is some $y \in X$ such that $(x, y) \in A$ and $(y, z) \in B$. This composite of subsets of $X \times X$ is associative. Moreover $R \circ R = R$ if $R$ is an equivalence relation on some subset of $X$; thus $B \circ A \subset R_{\mathcal{F}} \circ R_{\mathcal{F}} = R_{\mathcal{F}}$ if $A, B \subset R_{\mathcal{F}}$. Now, for each finite atlas $\mathcal{U}$ of the foliated space $X$, let $R_{\mathcal{U}}$ be the set of pairs $(x, y) \in X \times X$ such that both $x$ and $y$ lie in some plaque of $\mathcal{U}$, and let $R_{\mathcal{U}}^{(n)}$ denote the $n$-fold composite $R_{\mathcal{U}}$ with itself for each $n \in \mathbb{Z}_+$. Since $R_{\mathcal{F}} = \bigcup_{n=1}^{\infty} R_{\mathcal{U}}^{(n)}$, a map $d_{\mathcal{U}}: R_{\mathcal{F}} \to [0, \infty)$ can be defined by

$$ d_{\mathcal{U}}(x, y) = \inf \left\{ n \in \mathbb{N} \mid (x, y) \in R_{\mathcal{U}}^{(n)} \right\}; $$
that is, $d_U(x, y)$ is the length of a minimal finite sequence of plaques of $\mathcal{U}$ so that the first one contains $x$, the last one contains $y$, and consecutive plaques have nonempty intersection. It is easy to check that $d_U$ is a coarse metric on $R_{\mathcal{F}}$, and the leaves with $d_U$ are equi-coarsely convex. In [?], a key rôle is played by a coarse metric on the holonomy covers of the leaves, which is defined in the same way as $d_U$.

**Lemma 3.36.** The identity maps on the leaves, endowed with the coarse metrics induced by two finite regular atlases, form a family of equi-large scale equivalences.

*Proof.* Let $\mathcal{U}$ and $\mathcal{V}$ be finite regular *** atlases of the foliated space $X$. We can assume that $\mathcal{V}$ is a regular refinement *** of $\mathcal{U}$ by ***. So $d_U \le d_V$. Moreover, by ***, there is some $a \in \mathbb{Z}_+$ such that every plaque of $\mathcal{U}$ is contained in the union of at most $a$ plaques of $\mathcal{V}$, yielding $d_U \le a d_V$. $\square$

According to Lemma 3.36 and ***, the equi-coarse quasi-isometry type *** of the leaves with $d_U$ is independent of the choice of $\mathcal{U}$, and thus will be called the *canonical equi-coarse quasi-isometry type* of the leaves. In particular, the coarse structure on the leaves induced by $d_U$ is independent of the choice of $\mathcal{U}$, and can be called the *canonical coarse structure* on the leaves.

**Definition 3.37.** A distance $d$ on $R_{\mathcal{F}}$ is said to be *good* when it induces the leaf uniformity on $X$ and the canonical equi-coarse quasi-isometry type on the leaves.

**Remark.** By the obvious version of Corollary 3.16 for coarse metric spaces, a distance $d$ on $R_{\mathcal{F}}$ is good if and only if it induces the leaf uniformity on $X$ and the canonical coarse structure on the leaves, and moreover the leaves with $d$ are equi-coarsely convex.

When $(X, \mathcal{F})$ is $C^1$, for any riemannian metric on the leaves, continuous on $X$, the corresponding length structure on the leaves induces a good distance on $R_{\mathcal{F}}$.

********** THE FOLLOWING PROPOSITION CAN BE WRITTEN BEFORE WITH A WEAKER CONDITION BECAUSE ITS PROOF ONLY USES THE CONDITION THAT $d$ IS COARSELY QUASI-SOMETRIC TO $d_U$ **************

**Proposition 3.38.** For $\mathcal{U}$, $\mathcal{H}$ and $E$ as above, if $d$ is a good distance on $R_{\mathcal{F}}$, then the above inclusions of $\mathcal{H}$-orbits with $d_E$ into leaves with $d$ form a family of equi-coarse quasi-isometry equivalences.

*Proof.* Since the identity maps on the leaves with $d$ and $d_U$ are equi-coarse quasi-isometries ***, it is enough to prove that $d_E - 1 \le d_U \le d_E + 1$ on $R_{\mathcal{H}}$.

Let $(x, y) \in R_{\mathcal{H}}$. If $x = y$, then $d_E(x, y) = 0$ and $d_U(x, y) = 1$, and thus the inequalities of the statement hold at $(x, y)$. Therefore we can assume $x \ne y$.

Let $d_E(x, y) = n > 0$. Then there are indices $i_0, \dots, i_n$ such that the composite $h_{i_{n-1}, i_n} \circ \dots \circ h_{i_0, i_1}$ is defined at $x$ and maps $x$ to $y$. So there is a plaque $P_k$ of each chart $(U_{i_k}, \phi_{i_k})$, $k \in \{0, \dots, n\}$, such that $x \in P_0$, $y \in P_n$ and $P_{k_1} \cap P_k \ne \emptyset$ for all $k \in \{1, \dots, n\}$. This yields $d_U(x, y) \le n + 1$.
Let $d_U(x, y) = m$. Then there a sequence of plaques $Q_1, \dots, Q_m$ of $\mathcal{U}$ so that $x \in Q_1$, $y \in Q_m$ and $Q_{\ell-1} \cap Q_{\ell} \neq \emptyset$ for all $\ell \in \{2, \dots, m\}$. There are indices $j_0, \dots, j_{m+1}$ such that each $Q_{\ell}$ is a plaque of the flow box $(U_{j_{\ell}}, \phi_{j_{\ell}})$ for $\ell \in \{1, \dots, m\}$, and the points $x$ and $y$ lie in plaques $Q_0$ and $Q_{m+1}$ of the flow boxes $(U_{j_0}, \phi_{j_0})$ and $(U_{j_{m+1}}, \phi_{j_{m+1}})$, respectively. Hence the composite $h_{j_{m},j_{m+1}} \circ \dots \circ h_{j_0,j_1}$ is defined at $x$ and maps $x$ to $y$. So $d_E(x, y) \le m+1$. $\square$

*********************

**Corollary 3.39.** The leaves are of equi-coarse bounded geometry *** with any good distance on $R_{\mathcal{F}}$.

*Proof.* Since $E$ is finite, the induced graph structure on each $\mathcal{H}$-orbit is of finite type, and thus the result follows from Proposition 3.38. $\square$

**Theorem 3.40.** For any compact foliated space $(X, \mathcal{F})$, there exists a good distance on $R_{\mathcal{F}}$.

*Proof.* Let $\mathcal{U} = \{U_i, \phi_i\}$ be a finite regular *** atlas of $(X, \mathcal{F})$, which is a shrinking of another finite regular atlas. Then $\phi_i(U_i) = D_i \times Z_i$, and let $p_i : U_i \to Z_i$ denote the corresponding projection. For each index $i$, let $R_i$ denote the equivalence relation on $U_i$ whose equivalence classes are the plaques of the flow box $(U_i, \phi_i)$. Notice that $R_i = (U_i \times U_i) \cap \cup_j R_j$ by the regularity of $\mathcal{U}$, obtaining the following.

**Claim 2.** Each $R_i$ is open in $\cup_j R_j$ with the topology induced by the ambient space topology.

For each $i$, let $d_i : R_i \to [0, \infty)$ be the distance whose restriction to each plaque of $(U_i, \phi_i)$ corresponds to the euclidean metric on $D_i$ via $\phi_i$; i.e., for each $z \in Z_i$, the restriction of $d_i$ to the plaque $p_i^{-1}(z)$ corresponds to the euclidean metric on $D_i$ by the homeomorphism

$$p_i^{-1}(z) \xrightarrow{\phi_i} D_i \times \{z\} \equiv D_i.$$

The following assertion is easy to prove.

**Claim 3.** The sets $\cup_i d_i([0, t])$, $t > 0$, form a base of neighbourhoods of $\Delta_X$ in $\cup_i R_i$.

Let $\lambda_i$ be a partition of unity of $X$ subordinated to the open covering $\{U_i\}$. Suppose that $i$ runs in an index set with $N$ elements. For any fixed real number $0 < r < \frac{1}{N}$ and each $i$, let $U'_i = \lambda_i^{-1}((r, 1])$. These sets form an open covering of $X$, which is a shrinking of $\{U_i\}$. Let also $\{U''_i\}$ be a shrinking of $\{U'_i\}$, and set

$$R'_i = R_i \cap (U'_i \times U'_i), \quad R''_i = R_i \cap (U''_i \times U''_i).$$

Let $\bar{d} : \cup_i R_i \to [0, \infty)$ be the map defined by

$$\bar{d}(x, y) = \sum_{(x,y) \in R_i} \lambda_i(x) \lambda_i(y) d_i(x, y).$$

Observe that

$$(3.12) \qquad (x, y) \in R'_i \implies \bar{d}(x, y) \ge r^2 d_i(x, y).$$
Let A be the set of pairs $(x, y) \in \bigcup_i R_i''$ such that

$$ \{x, y\} \cap \overline{U_j''} \neq \emptyset \implies \{x, y\} \subset U_j' $$

for all $j$; note that $\{x, y\} \subset U_j'$ implies $(x, y) \in R_j'$ for all $(x, y) \in \bigcup_i R_i$ by the regularity of $\mathcal{U}$. The set A is clearly symmetric; i.e., $(x, y) \in A$ if and only if $(y, x) \in A$.

**Claim 4.** The set A is open in $\bigcup_i U_i$ with the topology induced by the ambient space topology.

First, the set A is open in $\bigcup_i R_i''$ because

$$ A = \bigcup_i R_i'' \cap \bigcap_j \left( (U_j' \times U_j') \cup \left( (X \setminus \overline{U_j''}) \times (X \setminus \overline{U_j''}) \right) \right). $$

On the other hand, each $R_i''$ is clearly open in $R_i$; so $\bigcup_i R_i''$ is open in $\bigcup_i R_i$ by Claim 2. Therefore A is open in $\bigcup_i R_i$ as stated in Claim 4.

As a first consequence of Claim 4, for each $x \in X$, the set

$$ A(x) = \{y \in X \mid (x, y) \in A\} $$

is open in X with the leaf topology.

For $(x, y) \in R_{\mathcal{F}}$, let $S(x, y)$ be the set of arbitrarily large finite sequences of points in the same leaf, $(x_0, \dots, x_n)$, such that $x_0 = x$, $x_n = y$ and $(x_{k-1}, x_k) \in A$ for all $k \in \{1, \dots, n\}$. Since $S(x, y) \neq \emptyset$ for all $(x, y) \in R_{\mathcal{F}}$, a map $d: R_{\mathcal{F}} \to [0, \infty)$ can be defined by setting

$$ d(x, y) = \inf \left\{ \sum_{k=1}^{n} \bar{d}(x_{i_{k-1}}, x_{i_k}) \mid (x_0, \dots, x_n) \in S(x, y) \right\}. $$

This $d$ is a pseudodistance on $R_{\mathcal{F}}$ because the following holds for all points $x, y, z$ in the same leaf:

$$ \left. \begin{array}{l} (x, x) \in S(x, x), \\ (x_0, \dots, x_n) \in S(x, y) \implies (x_n, \dots, x_0) \in S(y, x), \\ (x_0, \dots, x_m) \in S(x, y) \\ (x_m, \dots, x_{m+n}) \in S(y, z) \end{array} \right\} \implies (x_0, \dots, x_{m+n}) \in S(x, z). $$

**Claim 5.** For any $(x, y) \in R_{\mathcal{F}}$, if $x \in U_i''$, then

$$ d(x, y) \ge \begin{cases} r^2 \min\{d_i(x, y), d_i(x, U_i' \setminus U_i'')\} & \text{if } (x, y) \in R_i', \\ r^2 d_i(x, U_i' \setminus U_i'') & \text{if } (x, y) \notin R_i'. \end{cases} $$

To prove this assertion, let $(x_0, \dots, x_n) \in S(x, y)$. On the one hand, if $(x_{k-1}, x_k) \in R_i''$ for all $k \in \{1, \dots, n\}$, then

$$ \sum_{k=1}^{n} \bar{d}(x_{k-1}, x_k) \ge r^2 \sum_{k=1}^{n} d_i(x_{k-1}, x_k) \ge r^2 d_i(x_0, x_n) = r^2 d_i(x, y) $$
by (3.12). On the other hand, if there is some $k \in \{1, \dots, n\}$ with $(x_{k-1}, x_k) \notin R_i'',$ let

$$n_0 = \min \{k \in \{1, \dots, n\} \mid (x_{k-1}, x_k) \notin R_i''\}.$$

Since $x_{n_0-1} \in U_i''$, it follows that $(x_{n_0-1}, x_{n_0}) \in R_i'$ because $(x_{n_0-1}, x_{n_0}) \in A$. So

$$\sum_{k=1}^{n} \bar{d}(x_{k-1}, x_k) \geq \sum_{k=1}^{n_0} \bar{d}(x_{k-1}, x_k) \geq r^2 \sum_{k=1}^{n_0} d_i(x_{k-1}, x_k) \geq r^2 d_i(x_0, x_{n_0}) \geq r^2 d_i(x, U_i' \setminus U_i'')$$

by (3.12), which completes the proof of Claim 5.

To show that $d$ is indeed a distance, suppose that $d(x, y) = 0$ for some $(x, y) \in R_X$, and let us prove that $x = y$. Then $x \in U_i''$ for some $i$ and, since $d_i(x, U_i' \setminus U_i'') > 0$, it follows from Claim 5 that $(x, y) \in R_i''$ and $d_i(x, y) \leq d(x, y)/r^2 = 0$, yielding $x = y$ as desired because $d_i$ is a metric on $R_i$.

**Claim 6.** The restriction $d: \bigcup_i R_i \to [0, \infty)$ is continuous on the points of $\Delta_X$.

By Claim 2, it is enough to prove that each restriction $d: R_i \to [0, \infty)$ is continuous on the points of $\Delta_{U_i}$. Let $(x_n, y_n)$ be a sequence in $R_i$ which converges to some $(x, x) \in \Delta_{U_i}$ with respect to the topology on $R_i$ induced by the ambient space topology. Then $(x_n, y_n) \in A$ for $n$ large enough by Claim 4, and thus $(x_n, y_n) \in S(x_n, y_n)$, yielding $d(x_n, y_n) \leq \bar{d}(x_n, y_n)$, which converges to zero as $n \to \infty$ because $\bar{d}$ is continuous on $R_i$ with the topology induced by the ambient space topology. Hence $d(x_n, y_n) \to 0$, which shows Claim 6.

**Claim 7.** The sets $d^{-1}([0, t]) \cap \bigcup_i R_i$ form a base of neighbourhoods of $\Delta_X$ in $\bigcup_i R_i$.

By Claim 6, each set $d^{-1}([0, t]) \cap \bigcup_i R_i$ is a neighbourhood of $\Delta_X$ in $\bigcup_i R_i$. On the other hand, we get

$$d^{-1}([0, t]) \cap \bigcup_{i} R_{i}'' \subset \bigcup_{i} d_{i}^{-1}([0, t/r^{2}])$$

by Claim 5. Therefore Claim 7 follows from Claim 3.

As above, for any other regular *** atlas of $(X, \mathcal{F})$, $\mathcal{V} = \{(V_\alpha, \psi_\alpha)\}$ and each index $\alpha$, let $R_\alpha$ denote the equivalence relation on $V_\alpha$ whose equivalence classes are the plaques of $(V_\alpha, \psi_\alpha)$. The following assertion is obvious.

**Claim 8.** The sets $\bigcup_\alpha R_\alpha$ form a base of neighbourhoods of $\Delta_X$ in $\bigcup_i R_i$ when $\mathcal{V}$ runs over all finite regular refinements *** of $\mathcal{U}$.

By Claims 7 and 8, we easily get that $d$ induces the leaf uniformity.

It also follows from Claims 7 and 8 that $\bigcup_\alpha R_\alpha \subset d^{-1}([0, t])$ for some regular refinement $\mathcal{V} = \{(V_\alpha, \psi_\alpha)\}$ of $\mathcal{U}$ by Claim 8. This means that the $d$-diameter of each plaque of $\mathcal{V}$ is $\leq 1$. But, by ***, each plaque of $\mathcal{U}$ is contained in the union of at most $a$ plaques of $\mathcal{V}$ for some $a \in \mathbb{Z}_+$. So the $d$-diameter of every plaque of $\mathcal{U}$ is $\leq a$, yielding

$$d \leq a d_{\mathcal{U}}.$$

**Claim 9.** The map $d$ is bounded on $A$.
For all $(x, y) \in A$, we have $(x, y) \in S(x, y)$, yielding $d(x, y) \le \bar{d}(x, y)$. So Claim 9 follows because $\bar{d}$ is continuous and $\bigcup_i R_i''$ has compact closure in $\bigcup_i U_i$ with the topology induced by the ambient space topology.

By the definition of $d$, for any $\varepsilon > 0$ and each $(x, y) \in R_{\mathcal{F}}$, there is some $(x_0, \dots, x_n) \in S(x, y)$ such that

$$ \sum_{k=1}^{n} d(x_{k-1}, x_k) \le d(x, y) + \varepsilon; $$

Assuming $x \ne y$, we can suppose $x_{k-1} \ne x_k$ for all $k \in \{1, \dots, n\}$, and thus

$$ d_{\mathcal{U}}(x_{k-1}, x_k) = 1 \le \frac{1}{c} d(x_{k-1}, x_k), $$

where $c > 0$ is an upper bound of $d$ on $A$. Hence

$$ d_{\mathcal{U}}(x, y) \le \sum_{k=1}^{n} d_{\mathcal{U}}(x_{k-1}, x_k) \le \frac{1}{c} \sum_{k=1}^{n} d(x_{k-1}, x_k) \le \frac{1}{c} d(x, y) + \frac{\varepsilon}{c}, $$

yielding

$$ (3.14) \qquad d_{\mathcal{U}} \le \max \left\{ \frac{1}{c} d, 1 \right\} $$

on $R_{\mathcal{F}}$. From (3.13) and (3.14), the identity maps on the leaves with $d$ and $d_{\mathcal{U}}$ form a family of equi-large scale equivalences, which completes the proof. $\square$

## 3.8 Good length structures

*********************

**Question.** Does there exist a good distance on $R_{\mathcal{F}}$ induced by a length structure?

An affirmative answer to this question could be given by using Theorem 3.18 to prove that the distance constructed in the above proof is induced by a length structure. *** [Bing] ***

## 3.9 Coarse quasi-isometry type of leaves

The leaves of a compact foliated space are coarsely quasi-isometric to graphs of finite type by Theorem ???. The following result shows that this property is indeed a characterization of the coarse quasi-isometric type of leaves of compact foliated spaces, and of closed foliated manifolds too.

**Theorem 3.41.** Any connected graph of finite type is coarsely quasi-isometric to a leaf of a foliation by surfaces on a closed four manifold.
A finite connected graph is coarsely quasi-isometric to a compact manifold, and thus the second part of this theorem is trivial in this case. When the graph is infinite, the following well-known definitions are needed for the proof.

Let $G$ be a finitely generated group, and let $S$ be a finite symmetric system of generators of $G$. For each $g \in G$, denote by $|g|$ the length of a minimal expression of $g$ in terms of the elements of $S$. Then there is a distance on $G$ defined by letting

$$d(g_1, g_2) = |g_1 g_2^{-1}|.$$

In other words, this is the canonical metric of the Cayley graph of $G$ determined by $S$. This is the only right invariant metric on $G$ so that the distance between the identity element and any element of $S$ is one.

With more generality, if $H$ is a subgroup of $G$, then the space of right cosets $G/H$ has an induced distance defined by

$$d(g_1H, g_2H) = \inf\{|g| \mid g \in G, gg_1H = g_2H\}$$

for $g_1, g_2 \in G$. Alternatively, as right multiplication by $H$ is an isometry, this is also the distance on the space $G/H$ induced from $G$. Such metric spaces can be called homogeneous metric spaces of finitely generated groups. Different choices of finite symmetric systems of generators of $G$ give rise to lipschitz equivalent metrics on $G$, which thus induce lipschitz equivalent metrics on $G/H$.

Let now $G$ act on a set $X$ on the left. Then the distance on $G$ induces one on each orbit defined by

$$d(x, y) = \min\{|g| \mid g \in G, gx = y\}$$

for $x, y$ in $X$ in the same orbit. This is the metric induced by the connected graph of finite type whose set of vertices is $X$ and with an edge between different points $x, y$ if there is some $s \in S$ such that $sx = y$. For each $x \in X$, the mapping $gx \mapsto gG_x$ defines an isometry between the orbit $G(x)$ and the homogeneous metric space $G/G_x$, where $G_x$ is the stabilizer of $x$ under $G$. Any subgroup $H \subset G$ can be realized as a stabilizer $G_x$ for the $G$-action on $X = G/H$ induced by left multiplication, and $x = H$.

**Lemma 3.42.** A metric space is induced by connected graph of finite type if and only if it is isometric to a homogeneous metric space of a finitely generated group.

*Proof.* We only have to prove the “only if” part. Let $X$ be the set of vertices of a connected graph of finite type. Let $n$ be an upper bound for the number of edges that meet at every vertex, and let $G$ be the free group generated by a set $S$ with $n + 1$ generators. It is easy to check that it is possible to assign an element of $S$ to each edge of the graph so that there are no pairs of consecutive edges with the same assignment. Then a transitive action of $G$ on $X$ is well defined by setting

$$s(x) = \begin{cases} y & \text{if } s \text{ is assigned to some edge with vertices } x, y, \\ x & \text{otherwise,} \end{cases}$$
for each $s \in S$ and $x \in X$. This action obviously induces the given graph structure, and the result follows. □

**Remark.** The simple construction of the above proof is in the same spirit as that of C.C. Moore and J. Feldman [?]. They prove that a discrete equivalence relation on a borel space is generated by the action of a countable group. The group that they obtain is infinitely generated.

To finish the proof of Theorem 3.41, by Lemma 3.42, it is enough to show that, given any finitely generated group $G$ and any subgroup $H$ of infinite index in $G$, the infinite homogeneous metric space $G/H$ is coarsely quasi-isometric to a leaf of some foliation of dimension two on a closed manifold of dimension four. To begin with, let us find an action of $G$ on a compact metrizable space having a point whose stabilizer is precisely $H$. There are several ways of achieving that.

The first construction covers all possibilities. Let $Z$ be the space of maps from $G/H$ onto $\{0,1\}$ with the topology of pointwise convergence. Let $G$ act on $Z$ by $gx(rH) = x(g^{-1}rH)$ for $g \in G$ and $x \in Z$. Then the stabilizer of the characteristic function of the element $H$ of $G/H$ is precisely $H$. This action may not be considered to be very good, for although it has an invariant ergodic measure, it also has periodic orbits, and many other orbits which are of the form $G/K$, where $K$ is a proper subgroup of $H$.

If some more information about $G$ and $H$ is available, then better actions can be constructed. For example, if $G$ is a hyperbolic (non-elementary) group, then it acts naturally on its visual boundary $Z$. Each element of $G$ has at most two fixed points in $Z$, so that $G$ acts freely on a dense $G_\delta$ subset of $Z$ when this $Z$ is uncountable (which is the case if $G$ in non-elementary). So this action is valid for our purposes when $H$ is trivial.

The next construction is suggested by the theory of residually finite groups. Suppose that the subgroup $H$ of $G$ is the intersection of finite index subgroups of $G$. Let $\mathcal{H}$ denote this family of subgroups, and construct the inverse limit $Z$ of the finite discrete spaces $G/K$, with $K \in \mathcal{H}$. The action of $G$ on each $G/K$ is compatible with the bonding maps thus providing an action of $G$ on $Z$. The space $Z$ has a distinguished point, that which sits on top of the distinguished points of the homogeneous spaces $G/K$. Its stabilizer is precisely $\cap_{\mathcal{H}} K = H$. This action has an ergodic invariant measure.

If the subgroup $H$ was normal in $G$ and the quotient group $G/H$ was residually finite, the same construction applied to $G/H$ produces a compact group $Z$ containing $G/H$ as a dense subgroup. This compact group has a haar measure which is invariant.

In all these examples, the space $Z$ is a cantor set since $G/H$ is infinite. So a foliated space would be produced by suspension, not a manifold. To solve this problem, we can assume that $G$ is free, and consider any embedding of $Z$ in the 2-sphere $S^2$. A classical theorem in planar topology (to be found in Moise [19], for instance) says that if $f$ is a homeomorphism of a cantor set in the 2-sphere, then $f$ admits an extension to a homeomorphism of the sphere. So, since $G$ is free, the $G$-action on $Z$ extends to a $G$-action on $S^2$; i.e., a representation $G \to \mathrm{Homeo}(S^2)$. By taking a surface $R$ of appropriate genus, there is a representation $\pi_1(R) \to \mathrm{Homeo}(S^2)$ that factors through the representation of $G$, and thus its suspension is a foliation by surfaces of a closed four manifold (a foliated sphere bundle over a closed
surface) having leaves quasi-isometric to the given homogeneous metric space $G/H$.
# Chapter 4

# Topological Dynamics

## 4.1 Topological dynamics

A polish space is a topological space whose topology is second countable, and is induced by a complete metric.

The following terminology regarding subsets of a polish space $X$ will be frequently used. A good reference for all this and related material is Kechris [17].

**Definition 4.1.** A subset $A$ of a polish space $X$ is called

(1) *residual* if $A$ contains the intersection of a countable family of open dense subsets of $X$;

(2) *nowhere dense* if $A^-$ has empty interior;

(3) *meager* if $A$ is a countable union of nowhere dense sets;

(4) *borel* if $A$ is a member of the $\sigma$-algebra generated by the open subsets of $X$,

(5) *baire* (has the *baire property*) if there is an open set $U$ in $X$ such that the symmetric difference $A\Delta U = (A \setminus U) \cup (U \setminus A)$ is meager; and

(6) *analytic* if $A$ is the continuous image of a polish space.

The family borel subsets of $X$ forms a $\sigma$-algebra, which is the smallest one containing all open sets. The baire subsets of $X$ also form a $\sigma$-algebra, the smallest one containing all the open sets and all the meager sets; in particular, every borel set is a baire set. It is a theorem that every borel set is an analytic set, and every analytic set is a baire set. An advantage of using the class of analytic sets is that it is preserved by direct and inverse images under continuous maps.

The Kuratowski-Ulam theorem is the topological analog to Fubini's theorem. It is stated here for future reference; a proof can be found in [17].

**Theorem 4.2.** Let $A$ be a baire subset of a product $X \times Y$ of polish spaces. Then
1. For a residual set of $x \in X$, the set $A_x = \{y \in X \mid (x, y) \in A\}$ is baire.

2. A is meager (respectively, residual) if and only if for a residual set of points $x \in X$, the set $A_x$ is meager (respectively, residual).

**Definition 4.3.** A property $\mathcal{P}$ which points of a space $X$ may or may not have is said to be *generic* if the collection of those points of $X$ having said property $\mathcal{P}$ is a residual subset of $X$. In particular, a generic subset of a polish space is dense, hence non-empty.

**Definition 4.4.** Let $\mathcal{H}$ be a pseudogroup of local transformations of a polish space $Z$. A subset of $Z$ is said to be $\mathcal{H}$-saturated (or saturated) if it is a union of orbits of $\mathcal{H}$. The saturation of a subset $A \subset Z$, denoted by $\mathcal{H}(A)$, is the union of all orbits which meet $A$.

The saturations of open sets are open. Moreover, for $A \subset B \subset Z$, $\mathcal{H}(A)$ is dense in $\mathcal{H}(B)$ if $A$ is dense in $B$; hence $\mathcal{H}(A)$ is residual in $\mathcal{H}(B)$ if $A$ is residual in $B$. The orbits of $\mathcal{H}$ are the classes of an equivalence relation $R_{\mathcal{H}}$ on $Z$. Other properties which will be frequently used are listed in the following lemma.

**Lemma 4.5.** Suppose that $\mathcal{H}$ is countably generated.

1. The relation set $R_{\mathcal{H}}$ is an $F_\sigma$-subset of $Z \times Z$.

2. The saturation of a borel, analytic, baire or meager subset is a borel, analytic baire or meager subset, respectively.

3. If a set $A$ is residual in some open subset $V$ of $Z$, then $\mathcal{H}(A)$ is residual in $\mathcal{H}(V)$.

*Proof.* Let $S$ be a countable symmetric set of generators of $\mathcal{H}$, and let $S'$ be the set of all composites of elements of $S$, wherever defined. Then any transformation in $\mathcal{H}$ can be expressed as a combination of restrictions to open sets of transformations in $S'$.

The relation set $R_{\mathcal{H}}$ is the set of pairs $(x, y)$ with $x$ and $y$ in the same orbit. This is equal to the union of the graphs of maps in $S'$, which are easily seen to be $F_\sigma$-sets. Hence $R_{\mathcal{H}}$ is an $F_\sigma$-set because $S'$ is countable.

The properties about saturations follow from the equality

$$ \mathcal{H}(A) = \bigcup_{h \in S'} h(A \cap \operatorname{dom} h) $$

for any $A \subset Z$.

<figure><img src="image_7.png" /></figure>

**Definition 4.6.** The pseudogroup $\mathcal{H}$ is said to be *transitive* if it has a dense orbit. It is said to be *minimal* if every orbit is dense. A *minimal set* is a non-empty, saturated and closed subset of $Z$ which is minimal among the sets with these properties.

The following consequence is classical, (due to Birkhoff?).

**Proposition 4.7.** The union of those orbits of $\mathcal{H}$ which are dense in $Z$ is a $G_\delta$-set. In particular, this set is residual if and only if $\mathcal{H}$ is transitive.
*Proof.* Let {$U_n$} be a countable base for the topology of $Z$. Then the union of dense orbits is equal to the intersection of the saturations $\mathcal{H}(U_n)$, which are open in $Z$. $\square$

**Corollary 4.8.** If $\mathcal{H}$ is transitive, then the union of all the proper minimal sets is meager.

The following result is the topological zero-one law, a topological version of ergodicity.

**Theorem 4.9.** With the above notation, suppose that $\mathcal{H}$ is transitive. Then any saturated baire subset of $Z$ is either meager or residual.

*Proof.* Let $A$ be a saturated baire subset of $Z$. There is an open set $U$ such that the symmetric difference $A\Delta U$ is a meager set. If $A$ is not meager, then $U$ is non-empty and $U\setminus A$ is meager. Thus, if $A$ is neither meager nor residual, then there are non-empty open subsets $U$ and $V$ so that $A\cap U$ is residual in $U$ and $V\setminus A$ is residual in $V$. Hence the saturations $\mathcal{H}(A\cap U)$ and $\mathcal{H}(V\setminus A)$ are residual in $\mathcal{H}(U)$ and $\mathcal{H}(V)$, respectively. Since there is a dense orbit, the non-empty open sets $\mathcal{H}(U)$ and $\mathcal{H}(V)$ intersect in a non-empty set, and thus $\mathcal{H}(A\cap U)$ meets $\mathcal{H}(V\setminus A)$. But $\mathcal{H}(A\cap U) \subset A$ and $\mathcal{H}(V\setminus A) \subset Z\setminus A$, a contradiction. $\square$

The groupoid of germs of $\mathcal{H}$ is the topological groupoid of germs of maps in $\mathcal{H}$ with the operation induced by the composition of maps and the étale topology (see [?]). Its subspace of units can be canonically identified with $Z$. For each $x \in Z$, the group of elements of this groupoid whose source and range is $x$ will be called the *group of germs of $\mathcal{H}$* at $x$. If $x$ and $y$ are points in the same orbit, then an isomorfism between the corresponding groups of germs is defined by conjugation by any element of this groupoid with source $x$ and range $y$. Thus the group of germs of each orbit is well defined up to conjugation in the groupoid of germs of $\mathcal{H}$.

A result of basic importance for the contents of this paper is the following theorem, due independently to G. Hector [16] and to D. Epstein, K. Millet and D. Tischler [9].

**Theorem 4.10.** With the above notation, if $\mathcal{H}$ is countably generated, then the union of orbits with trivial groups of germs is a dense $G_\delta$ subset of $X$, hence borel and residual.

**Corollary 4.11.** If $\mathcal{H}$ is transitive and countably generated, then there is a residual saturated subset of $x$ consisting of dense orbits with trivial groups of germs.

The following shows that, for recurrent systems of compact generation of minimal pseudogroups, recurrency holds indeed in a baire sense.

**Theorem 4.12.** Let $\mathcal{H}$ be a compactly generated minimal pseudogroup of local transformations of a locally compact space $Z$, let $U$ be a relatively compact open subset of $Z$, let $\mathcal{G}$ denote the restriction of $\mathcal{H}$ to $U$, and let $E$ be recurrent symmetric system of compact generation of $\mathcal{H}$ on $U$. Then, for any baire subset $B$ of $U$, either $\mathcal{G}(B)$ is meager or else the following property holds: There is a residual $\mathcal{G}$-saturated subset $Y$ of $U$ and there is some $R > 0$ such that $\emptyset \cap B$ is an $R$-net in $(\emptyset, d_E)$ for any $\mathcal{G}$-orbit $\emptyset$ in $Y$.
*Proof.* The set $B$ is not meager by Lemma 4.5. Then there is some non-empty open subset $V$ of $U$ so that $V \setminus B$ is meager in $V$. Hence $\mathcal{G}(V \setminus B)$ is meager in $U$ by Lemma 4.5, and thus the complement $Y = U \setminus \mathcal{G}(V \setminus B)$ is residual in $U$ and $\mathcal{G}$-saturated. Since $\mathcal{H}$ is minimal and $E$ recurrent, there is some $R > 0$ such that $\mathcal{O} \cap V$ is an $R$-net in $(\mathcal{O}, d_E)$ for any $\mathcal{G}$-orbit $\mathcal{O}$ (Lemma 3.23); but $\mathcal{O} \cap V \subset B$ if $\mathcal{O} \subset Y$, and the result follows. $\square$

The following example shows that there is no measure theoretic counterpart of Theorem 4.12.

**Example 4.13.** Let $\mathcal{H}$ be the minimal pseudogroup generated by some rotation $h$ of the circle $S^1$ with dense orbits. Then $\mathcal{H}$ satisfies the conditions of Theorem 4.12 and $E = \{h, h^{-1}\}$ is a recurrent system of compact generation of $\mathcal{H}$ on $S^1$. For each positive integer $n$, let $I_n$ be a non-empty open arc in $S^1$ whose length is $< 1/n2^n$. Then

$$A = \bigcup_{n=1}^{\infty} \bigcup_{i=-n}^{n} h^i(I_n)$$

is a borel set whose lebesgue measure is smaller than the length of $S^1$. So its complement $B = S^1 \setminus A$ is a borel set of positive lebesgue measure, and thus the saturation $\mathcal{H}(B)$ is of full measure (see [?]). Nevertheless, for each $n$, every orbit $\mathcal{O}$ meets $I_n$ at some point $x$, and thus $\mathcal{O} \cap A$ the points $h^i(x)$ for $-n \le i \le n$, which form a ball of radius $n+1$ in $(\mathcal{O}, d_E)$. Hence $\mathcal{O} \cap B$ is not a net in $(\mathcal{O}, d_E)$ for any orbit $\mathcal{O}$.

**Definition 4.14.** Let $(X, \mathcal{F})$ be a foliated space. A subset of $X$ is said to be *saturated* if it is a union of leaves. If $A \subset X$, the saturation of $A$, denoted by $\mathcal{F}(A)$, is the union of all leaves which meet $A$.

If $\mathcal{H}$ is any representative of the holonomy pseudogroup of $\mathcal{F}$, acting on a locally compact polish space $Z$, then there is a homeomorphism between the space of leaves $X/\mathcal{F}$ and the space of orbits $Z/\mathcal{H}$. This induces a bijection between saturated subsets of $X$ and saturated subsets of $Z$, which preserves open and closed sets, $G_\delta$ and $F_\sigma$ sets, borel sets, residual and meager sets, baire sets and analytic sets. This can be easily seen by taking the representative of the holonomy pseudogroup induced by any defining cocycle of $\mathcal{F}$.

Let $R_{\mathcal{F}}$ be the equivalence relation on $X$ whose classes are the leaves of $\mathcal{F}$. Each leaf is a borel subset of $X$. The behaviour of the saturation is worse in foliated spaces than in pseudogroups: it is well known and easy to prove that the saturation of an open set is open, but the saturation of a meager, borel or baire set may not be meager, borel or baire, respectively. Nevertheless the following holds.

**Lemma 4.15.** 1. The relation set $R_{\mathcal{F}}$ is an $F_\sigma$-subset of $X \times X$.

2. The saturation of an analytic set is analytic.

3. If a set $A$ is residual in some open subset $V$ of $X$, then $\mathcal{F}(A)$ is residual in $\mathcal{F}(V)$.
*Proof.* Let $(U_i, p_i, h_{i,j})$ be a countable defining cocycle of $\mathcal{F}$ with $p_i: U_i \to Z_i$, and let $\mathcal{H}$ be the representative of the holonomy pseudogroup generated by the local transformations $h_{i,j}$ of $Z = \bigsqcup_i Z_i$. Then

$$R_{\mathcal{F}} = \bigsqcup_{i,j} (p_i \times p_j)^{-1} (R_{\mathcal{H}} \cap (Z_i \times Z_j)),$$

which is an $F_\sigma$-subset of $X \times X$ by Lemma 4.5 since each $p_i \times p_j$ is a trivial fiber bundle with $\sigma$-compact fibers.

The relation between saturations in $X$ and $Z$ is the following:

$$\mathcal{F}(A) = \bigsqcup_{i,j} p_i^{-1}(\mathcal{H}(p_j(A \cap U_j)) \cap Z_i)$$

for any $A \subset X$. Hence the class of analytic sets is preserved by saturations by Lemma 4.5, since the property of being analytic is preserved by direct and inverse images under continuous maps.

The last part of the statement follows similarly.

<figure><img src="image_8.png" /></figure>

There are other saturations of a set which are relevant because they preserve the property of being borel. They are well known in descriptive set theory, and they will prove useful here.

**Definition 4.16.** Let $A$ be a subset of the foliated space $(X, \mathcal{F})$. The *Vaught transforms* of $A$ are the sets $\mathcal{F}_v(A)$ and $\mathcal{F}^v(A)$ defined as follows: $\mathcal{F}_v(A)$ is the union of all leaves $L$ such that the intersection $L \cap A$ is non-meager in $L$, and $\mathcal{F}^v(A)$ is the union all of such leaves for which $L \cap A$ is residual in $L$.

The following fact will be needed.

**Lemma 4.17.** Let $A$ be a meager subset of $X$. Then the Vaught saturation $\mathcal{F}_v(A)$ is meager.

*Proof.* In general, if $A = \bigsqcup A_n$ is a countable union, then $\mathcal{F}_v(A) = \bigsqcup_n \mathcal{F}_v(A_n)$, simply because $L \cap A = \bigsqcup_n (L \cap A_n)$ for any leaf $L$. Therefore, since $X$ can be covered by countable many flow boxes of the form $Z \times B$, it will be enough to consider the case in which $A \subset Z \times B$. Then the Vaught transform of this set $A$ is the saturation of the set of plaques $P$ such that $P \cap A$ is non-meager in $P$. But $A$ being meager, the intersection $P \cap A$ is meager for all plaques $P$ except a meager set of them (Theorem 4.2). □

**Definition 4.18.** A foliated space is said to be *transitive* if it has a dense leaf. It is said to be *minimal* if every leaf is dense.

The following results for foliated spaces are versions (and easy consequences) of the corresponding results for pseudogroups.

**Proposition 4.19.** The union of those leaves of $\mathcal{F}$ which are dense in $X$ is a $G_\delta$-set. In particular, this set is dense if and only if $\mathcal{F}$ is transitive.

**Corollary 4.20.** In a transitive foliated space, the union of all the proper minimal sets is meager.
**Theorem 4.21.** With the above notation, suppose that $\mathcal{F}$ is transitive. Then any baire saturated subset of $X$ is either meager or residual.

**Theorem 4.22.** With the above notation, the union of leaves which have no holonomy is a dense $G_\delta$ subset of $X$, hence borel and residual.

**Corollary 4.23.** In a transitive foliated space there is a residual saturated set consisting of leaves which are both dense and without holonomy.

**Definition 4.24.** A measurable space $Y$ is countable separated if there is a countable collection $\{B_n\}$ of measurable subsets of $Y$ such that any $y \in Y$ is the intersection of all the $B_n$ in the collection containing $y$.

**Example 4.25.** Let $Y$ be a first countable topological space that satisfies the $T_0$-separation axiom. Then the borel structure on $Y$ is countable separated.

**Proposition 4.26.** Let $X$ be a transitive foliated space, $Y$ a countable separated measurable space. If $f: X \to Y$ is a measurable map which is constant along the leaves of $X$, then there is a residual saturated set in $X$ where $f$ is constant.

*Proof.* Let $B_n$ be the family of subsets of $Y$, and assume that $B_n$ is closed under complements. For each $n$, $f^{-1}B_n$ is a Borel saturated subset of $X$, hence either it is residual or its complement is residual. Let $B = \bigcap \{B_n \mid f^{-1}B_n \text{ is residual in } X\}$. Then $B$ contains exactly one point. Indeed, it is non-empty, and if $x$ and $y$ are distinct points in $B$, then there is $B_n$ which contains $x$ but not $y$, contradicting the definition of $B$. $\square$

The following foliated version of recurrence (on open sets) follows from Lemma 3.23.

**Lemma 4.27.** If $X$ is compact, then, for any open subset $V$ of $X$ that meets every leaf, there is some $K > 0$ such that $L \cap V$ is a $K$-net in $L$ for every leaf $L$ of $\mathcal{F}$.

This result also follows by observing that the mapping $x \mapsto d_{L_x}(x, L_x \cap V)$ is upper semicontinuous on $X$, and thus it reaches a maximum on $X$; here, $L_x$ denotes the leaf that contains the point $x$, and $d_{L_x}$ is the riemannin distance on $L_x$.

Another proof of Lemma 4.27 can be given by assuming the existence of a sequence of balls in the leaves, with divergent sequence of radius, which do not meet the open set $V$. Then the limit set in $X$ of such a sequence do not meet $V$ as well. But this limit set is a non-empty closed foliated subspace, and thus $V$ does not meet all leaves, which is a contradiction.

About recurrence on baire sets, there is no exact version for foliated spaces of Theorem 4.12. In fact, any minimal compact foliated space $(X, \mathcal{F})$ of positive dimension contains a baire set $B$ whose saturation is the whole of $X$ and which meets every leaf at one point. This set $B$ can be defined by picking up one point in the intersection of any small enough local transversal $Z$ with every leaf ($Z$ meets every leaf by minimality). This $Z$ may be assumed to be meager by Theorem 4.2 since $\mathcal{F}$ is of positive dimension, and thus $B$ is meager too. Thus the singleton $L \cap B$ can not be a net in any unbounded leaf $L$.

Two possible changes can be introduced to adapt Theorem 4.12 to foliated spaces: either analytic sets are considered instead of baire sets, or a Vaught transform is used instead of the usual saturation.
**Theorem 4.28.** Let $(X, \mathcal{F})$ be a compact and minimal foliated space, and let $A$ be an analytic subset of $X$. Then, either $\mathcal{F}(A)$ is meager or else the following property holds: There is a residual saturated subset $Y$ of $X$ and there is some $R > 0$ such that the intersection $L \cap A$ is an $R$-net in $L$ for each leaf $L \subset Y$.

*Proof.* Let $(U_i, p_i, h_{i,j})$ be a finite defining cocycle of $\mathcal{F}$ with $p_i : U_i \to Z_i$, and let $\mathcal{H}$ be the representative of the holonomy pseudogroup generated by the local transformations $h_{i,j}$ of $Z = \bigsqcup_i Z_i$. Then the set $A' = \bigsqcup_i p_i(A \cap U_i)$ is an analytic subset of $Z$. Suppose that $(U_i, p_i, h_{i,j})$ is a shrinking of another defining cocycle; thus $\mathcal{H}$ is the restriction to $Z$ of a pseudogroup $\tilde{\mathcal{H}}$ on a polish space $\tilde{Z}$, where $Z$ is a relatively compact open subset that meets all orbits, and so that the local transformations $h_{i,j}$ form a recurrent system of compact generation of $\tilde{\mathcal{H}}$ on $Z$. Hence, by Theorem 4.12, either $\mathcal{H}(A')$ is meager, or else the following property holds: There is a residual $\mathcal{H}$-saturated subset $Y'$ and some $R' > 0$ such that $\emptyset \cap A'$ is an $R'$-net in $(\emptyset, d_E)$ for any $\mathcal{H}$-orbit $\emptyset$ in $Y'$. Then the result follows with the saturated subset $Y \subset X$ that corresponds to $Y'$, and with $R = R' + r$, where $r > 0$ is any upper bound of the diameter of the plaques of the above defining cocycle. $\square$

**Remark.** The above argument fails for baire sets because $A'$ may not be baire if $A$ is only known to be baire.

The following second version of foliated recurrency on baire sets follows with a sharpening of one the possible arguments indicated to prove foliated recurrency on open sets: by using the limit set of a sequence of balls in the leaves with divergent sequence of radius.

**Theorem 4.29.** Let $(X, \mathcal{F})$ be a compact minimal foliated space, and let $B$ be a baire subset of $X$. Then, either the Vaught transform $\mathcal{F}_v(B)$ of $B$ is meager or else the following property holds: There is a residual saturated subset $Y$ of $X$ and there is $R > 0$ such that the intersection $L \cap B$ is an $R$-net in $L$ for each leaf $L \subset Y$.

*Proof.* Suppose first that $B$ is not meager. Then there is a non-empty open set $U$ such that $U \setminus B$ is meager in $U$. By Lemma 4.17, the Vaught saturation $\mathcal{F}_v(U \setminus B)$ of $U \setminus B$ is meager in $X$; hence the complement $Y = X \setminus \mathcal{F}_v(U \setminus B)$ is residual and saturated. It also satisfies the following property: if $L$ is a leaf of $Y$ with $L \cap U \neq \emptyset$, then $L \cap B \neq \emptyset$. Indeed, such leaf $L$ must meet $U \setminus B$ in a meager set; since $L \cap U$ is open in $L$, the intersection $L \cap B$ cannot be empty.

It will now be shown that there is some $R > 0$ so that, for every leaf $L \subset Y$, $d_L(x, L \cap B) < R$ for all $x \in L$. If this was not the case, then for each $n = 1, 2, \dots$, there exits a ball $B(x_n, n)$ contained in a leaf $L_n \subset Y$ and disjoint from $B$. The set $\limsup_n B(x_n, n)$ is closed, non-empty and saturated, therefore it agrees with $X$ by minimality. This in turn implies that infinitely many balls $B(x_n, n)$ meet the open set $U$. Since these balls lie in balls contained in leaves of $Y$, they must also meet $B$, a contradiction.

In general, if $\mathcal{F}_v(B) = \bigcup B_n$, where $B_n$ consists of points at distance $\le n$ from a non-meager part of $B$, each $B_n$ is a baire set, and if $\mathcal{F}_v(B)$ is not meager, then one of the $B_n$ is also not meager. The previous argument applies to this $B_n$ and the desired conclusion follows. $\square$
**Corollary 4.30.** Let $B$ be a baire subset of $X$ which is not meager. Then for every leaf $L$ of a residual set, the intersection $L \cap B$ approaches all ends of $L$.

**Remark.** This result is a sort of topological analogue of the “Proposition fundamentelle” of E. Ghys [11]. The statement is false if the foliated space $X$ is only assumed to be transitive. The reason why it should fail will become apparent later. There is a similar statement available for transitive foliations; its formulation is left to the reader. This recurrence result will be amplified in ??.

## 4.2 A discrete version of local Reeb stability

Let $\mathcal{H}$ be a compactly generated minimal pseudogroup of local transformations of a locally compact space $Z$, let $U$ be a relatively compact open subset of $Z$, let $\mathcal{G}$ denote the restriction of $\mathcal{H}$ to $U$, and let $E$ be recurrent symmetric system of compact generation of $\mathcal{H}$ on $U$.

The following notation will be used. Let $E^m$ denote the $m$-fold cartesian product $E \times \cdots \times E$ for each $m \in \mathbb{Z}_+$, and set $E^0 = \{\mathrm{id}_Z\}$. For every $m \in \mathbb{Z}_+$ and $g = (g_1, \dots, g_m) \in E^m$, its *domain* is the set $\mathrm{dom}\,g = \mathrm{dom}(g_1 \circ \cdots \circ g_m)$, which may be empty; let $g(x) = g_1 \circ \cdots \circ g_m(x)$ for every $x \in \mathrm{dom}\,g$; and let $g^{-1} = (g_n^{-1}, \dots, g_1^{-1})$. For other $n \in \mathbb{Z}_+$ and $h = (h_1, \dots, h_n) \in E^n$, let $g \circ h = (g_1, \dots, g_m, h_1, \dots, h_n) \in E^{m+n}$. Finally, for $r \ge 1$, let $E^{<r}$ denote the union of the sets $E^m$ with $1 \le m < r$.

Let $U_0$ be the set of points in $U$ where $\mathcal{G}$ has trivial groups of germs, which is a dense $G_\delta$-subset of $U$ by Theorem 4.10.

**Lemma 4.31.** For every $r > 0$ and each $x \in U_0$, there is an open neighbourhood $V(x, r)$ of $x$ in $U$ such that:

* $V(x, r) \subset \mathrm{dom}\,g$ for every $g \in E^{<r}$ with $x \in \mathrm{dom}\,g$;

* there is a unique map $\phi_{x,y,r} : B_E(x,r) \to B_E(y,r)$ for every $y \in V(x,r)$ satisfying $\phi_{x,y,r}(g(x)) = g(y)$ for all $g \in E^{<r}$ with $x \in \mathrm{dom}\,g$;

* $d_E(\phi_{x,y,r}(z), \phi_{x,y,r}(z')) \le d_E(z, z')$ for all $r > 0$, and points $y \in V(x,r)$ and $z, z' \in B_E(x,r)$.

*Proof.* Since $x \in U_0$ and the sets $E^m$ are finite, there is some open neighbourhood $V(x, r)$ of $x$ in $U$ satisfying the first condition of the statement, and moreover so that, if $x \in \mathrm{dom}\,g$ and $g(x) = x$ for some $g \in E^{<4r}$, then $V(x, r) \subset \mathrm{dom}\,g$ and $g(y) = y$ for all $y \in V(x, r)$.

Fix any $y \in V(x, r)$. Each point in $B_E(x, r)$ is of the form $g(x)$ for some $g \in E^{<r}$ with $x \in \mathrm{dom}\,g$, and thus $y \in \mathrm{dom}\,g$ too. Suppose that $g(x) = h(x)$ for another $h \in E^{<r}$ with $x \in \mathrm{dom}\,h$. Then $h^{-1} \circ g \in E^{<2r}$, $x \in \mathrm{dom}(h^{-1} \circ g)$ and $h^{-1} \circ g(x) = x$. It follows that $y \in \mathrm{dom}(h^{-1} \circ g)$ and $h^{-1} \circ g(y) = y$; i.e., $g(y) = h(y)$. Therefore a map $\phi_{x,y,r} : B_E(x, r) \to B_E(y, r)$ is well defined by the assignment $g(x) \mapsto g(y)$ for $g \in E^{<r}$.

For $z, z' \in B_E(x, r)$, there are some $g, g' \in E^{<r}$ whose domains contain $x$ and such that $g(x) = z$ and $g'(x) = z'$. Let $m = d_E(z, z') < 2r$. There is some $h \in E^m$ such that $z \in \mathrm{dom}\,h$
and $h(z') = z$. Thus $h \circ g'(x) \in \text{dom } g^{-1}$ and $g^{-1} \circ h \circ g'(x) = x$ with $g^{-1} \circ h \circ g' \in E^{<4r}$. It follows that $y$ is in the domain of $g^{-1} \circ h \circ g'$ and $g^{-1} \circ h \circ g'(y) = y$. Hence $g'(y) \in \text{dom } h$ and $h \circ g'(y) = g(y)$, yielding $d_E(g(y), g'(y)) \le m$ as desired to prove the last assertion of the statement. $\square$

Let $U_E$ be the complement in $U$ of the $\mathcal{G}$-saturation of the union of boundaries in $U$ of the domains of the maps in $E$. Such a $U_E$ is a dense $G_\delta$ set, and thus so is $U_0 \cup U_E$. Note that $U_E$ does not meet the domain of any composite of maps in $E$. The following version for pseudogroups of the local Reeb stability is similar to the version given in [?].

**Proposition 4.32.** The open neighbourhoods $V(x,r)$ of Lemma 4.31 can be chosen so that the maps $\phi_{x,y,r}$ are isometric bijections for $r > 0$, and points $x \in U_0 \cap U_E$ and $y \in V(x,r)$.

*Proof.* For every $x \in U_0 \cap U_E$ and each $r > 0$, we can choose $V(x,r)$ satisfying the conditions of the proof of Lemma 4.31, and moreover so that:

* If $V(x,r)$ meets the domain of some $g \in E^{<r}$, then $x \in \text{dom } g$;

* if $g(y) = y$ for some $g \in E^{<4r}$ and $y \in V(x,r) \cap \text{dom } g$, then $x \in \text{dom } g$ and $g(x) = x$.

Fix any $y \in V(x,r)$. Any point in $B_E(y,r)$ is of the form $h(y)$ for some $h \in E^{<r}$ with $y \in \text{dom } h$. For such an $h$, we have $x \in \text{dom } h$, $h(x) \in B_E(x,r)$ and $\phi_{x,y,r}(h(x)) = h(y)$. Therefore $\phi_{x,y,r}$ is surjective.

If $z, z' \in B(x,r)$, then $z = g(x)$ and $z' = g'(x)$ for some $g, g' \in E^{<r}$ whose domains contain $x$. Hence any $y \in V(x,r)$ is in the domains of $g$ and $g'$, and so $\phi_{x,y,r}(z) = g(y)$ and $\phi_{x,y,r}(z') = g'(y)$. Set $m = d_E(g(y), g'(y)) < 2r$. There is some $h \in E^m$ so that $g'(y) \in \text{dom } h$ and $h \circ g'(y) = g(y)$. So $g^{-1} \circ h \circ g' \in E^{<4r}$, $y \in \text{dom}(g^{-1} \circ h \circ g')$ and $g^{-1} \circ h \circ g'(y) = y$. Therefore $x \in \text{dom}(g^{-1} \circ h \circ g')$ and $g^{-1} \circ h \circ g'(x) = x$. Thus $h(z') = z$, yielding $d_E(z, z') \le m$. It follows that $\phi_{x,y,r}$ is an isometric bijection by Lemma 4.31. $\square$

We also need the following weaker version of Reeb stability around arbitrary points in $U_0$.

**Proposition 4.33.** The open neighbourhoods $V(x,r)$ of Lemma 4.31 can be chosen so that the following properties are satisfied for some $C \ge 1$ and $K > 0$:

* The maps $\phi_{x,y,r}$, for $r > 0$, and points $x \in U_0$ and $y \in V(x,r)$, form a uniform family of non-expanding bi-lipschitz maps with bi-lipschitz distortion $C$;

* the set $B_E(y,r/C) \cap \text{im } \phi_{x,y,r}$ is a $K$-net in $(B_E(y,r/C), d_E)$ for all $r > 0$, and all points $x \in U_0$ and $y \in V(x,r)$.

*Proof.* By Theorem 3.26 (or Proposition 3.29) and Corollary 3.25, there is an extension $\bar{g}$ of each $g \in E$ with $\text{dom } \bar{g} \subset \text{dom } \bar{g}$ and such that $\bar{E} = \{\bar{g} \mid g \in E\}$ is a system of compact generation on some relatively compact open subset $U' \subset Z$ with $\bar{U} \subset U'$. For $m \in \mathbb{Z}_+$ and $g = (g_1, \dots, g_m) \in E^m$, let $\bar{g} = (\bar{g}_1, \dots, \bar{g}_m) \in \bar{E}^m$.

For every $x \in U_0$ and each $r > 0$, we can choose $V(x,r)$ satisfying the conditions of the proof of Lemma 4.31, and moreover so that: