* If $V(x, r)$ meets the domain of some $g \in E^{<4r}$, then $x \in \text{dom } \bar{g}$;

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

These new conditions can be assumed because $x \in U_0$, the sets $E^m$ are finite, and $\overline{\text{dom } g} \subset \text{dom } \bar{g}$ for each $g \in E^m$.

There is some $C \ge 1$ such that

$$ (4.1) \qquad d_{\bar{E}}(z, z') \le d_E(z, z') \le C d_{\bar{E}}(z, z') $$

for all $z, z' \in U$ lying in the same orbit; here, the first inequality holds because $\bar{g}$ is an extension of each $g \in E$, and the second inequality follows by Proposition 3.29.

Fix some $r > 0$, and points $x \in U_0$ and $y \in V(x, r)$. For any $z, z' \in B_E(x, r)$, we have $z = g(x)$ and $z' = g(x')$ for some $g, g' \in E^{<r}$. Hence $y \in \text{dom } g \cap \text{dom } g'$, $\phi_{x,y,r}(z) = g(y)$ and $\phi_{x,y,r}(z') = g'(y)$. Let $m = d_E(g(y), g'(y)) < 2r$. 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 $g^{-1} \circ h \circ g'(y) = y$. It follows that $x \in \text{dom}(\bar{g}^{-1} \circ \bar{h} \circ \bar{g}')$ and $\bar{g}^{-1} \circ \bar{h} \circ \bar{g}'(x) = x$, and thus $\bar{h}(z) = z'$ with $\bar{h} \in \bar{E}^m$. Therefore

$$ d_{\bar{E}}(z, z') \le d_E(\phi_{x,y,r}(z), \phi_{x,y,r}(z')), $$

yielding

$$ d_E(z, z') \le C d_E(\phi_{x,y,r}(z), \phi_{x,y,r}(z')) $$

by (4.1). Then the first part of the statement follows by Lemma 4.31.

Fix $r$ and $x$ as above. Since the maps in $\bar{E}$ are extensions of the maps in $E$, the set $V(x, r)$ satisfies the conditions of the proof of Lemma 4.31 with respect to $\bar{E}$ too. So the map $\bar{\phi}_{x,y,r}$, given by Lemma 4.31 with $\bar{E}$ instead of $E$, is defined for each $y \in V(x, r)$. Moreover $\bar{\phi}_{x,y,r}$ equals $\phi_{x,y,r}$ on $B_E(x, r)$, which is contained in $B_{\bar{E}}(x, r)$ by (4.1). Since $B_{\bar{E}}(x, r/C) \cap U \subset B_E(x, r)$ by (4.1), the map $\bar{\phi}_{x,y,r}$ equals $\phi_{x,y,r}$ on $B_{\bar{E}}(x, r/C) \cap U$ for any $y \in V(x, r)$, and thus

$$ (4.2) \qquad \bar{\phi}_{x,y,r}(B_{\bar{E}}(x, r/C) \cap U) \subset \text{im } \phi_{x,y,r}. $$

On the other hand, for any $z \in B_E(y, r/C)$, we have $z = g(y)$ for some $g \in E^{<r/C}$ with $y \in \text{dom } g$. Hence $x \in \text{dom } \bar{g}$ and $\bar{g}(x) \in B_{\bar{E}}(x, r/C)$ because $\bar{g} \in \bar{E}^{<r/C}$, and $\bar{\phi}_{x,y,r}(\bar{g}(x)) = z$. So

$$ (4.3) \qquad B_E(y, r/C) \subset \bar{\phi}_{x,y,r}(B_{\bar{E}}(x, r/C)). $$

By Lemma 3.23, there is some $R > 0$ so that $\emptyset \cap U$ is an $R$-net in $(\emptyset, d_{\bar{E}})$ for every $G'$-orbit $\emptyset$. Then $B_{\bar{E}}(x, r/C) \cap U$ is an $R$-net in $(B_{\bar{E}}(x, r/C), d_{\bar{E}})$, and thus $\bar{\phi}_{x,y,r}(B_{\bar{E}}(x, r/C) \cap U)$ is an $R$-net in $(\bar{\phi}_{x,y,r}(B_{\bar{E}}(x, r/C)), d_{\bar{E}})$ by Lemma 4.31 applied to $\bar{\phi}_{x,y,r}$. Therefore $B_E(y, r/C) \cap \text{im } \phi_{x,y,r}$ is a $CR$-net in $(B_E(y, r/C), d_E)$ by (4.1), (4.2) and (4.3), showing the last part of the statement. $\square$

**Remark.** *** *** *** version for orbits with finite groups of germs *** *** ***
# Chapter 5

## Generic coarse geometry of leaves

***

### 5.1 Coarsely quasi-isometric leaves

This section is devoted to the study of the equivalence relation on a compact foliated space, where two points are equivalent when they are in coarsely quasi-isometric leaves. Its equiv- alence classes will be shown to be baire sets. So, for transitive compact foliated spaces, the classes should be either meager or residual by Theorem 4.21; in fact, it will be shown that, if there is a residual class, then all dense leaves without holonomy are uniformly coarsely quasi- isometric to each other. For minimal compact foliated spaces, the existence of a residual class is characterized by a coarse property satisfied by any leaf without holonomy.

It is more convenient to begin with the pseudogroup version of the problem. So let $\mathcal{H}$ be a compactly generated pseudogroup of local transformations of a locally compact polish 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 a recurrent symmetric system of compact generation of $\mathcal{H}$ on $U$. In this section, all metric concepts and properties in the $\mathcal{G}$-orbits will be considered with respect to the metric $d_E$ induced by $E$; thus no further reference to this metric will be used. As in the above section, let $U_0$ be the union of $\mathcal{G}$-orbits with trivial groups of germs. Moreover let $\mathcal{G} \times \mathcal{G}$ be the pseudogroup of local transformations of $U \times U$ generated by the maps $h \times g$ with $h, g \in \mathcal{G}$.

**Theorem 5.1.** With the above notation, the equivalence relation "$x \sim y$ if and only if the orbits $\mathcal{G}(x)$ and $\mathcal{G}(y)$ are coarsely quasi-isometric" has a borel relation set in $U_0 \times U_0$, and has a baire relation set in $U \times U$; in particular, it has borel equivalence classes in $U_0$, and baire equivalence classes in $U$.

The following sets are introduced to prove this theorem. For integers $K, C \ge 1$, let $Y(K, C)$ be the set of pairs $(x, y) \in U_0 \times U_0$ such that there is a coarse quasi-isometry $f: A \to B$ between $\mathcal{G}(x)$ and $\mathcal{G}(y)$ with coarse distortion $(K, C)$, and so that $x \in A, y \in B$ and $f(x) = y$.
**Lemma 5.2.** For $K > 0$ and $C \ge 1$, there are some $K' \ge K$ and $C' \ge C$ such that $Y(K', C')$ contains the closure of $Y(K, C)$ in $U_0 \times U_0$.

*Proof.* Let $(x, y)$ be a pair in the closure of $Y(K, C)$ in $U_0 \times U_0$. For each positive integer $n$, let $V(x, n)$ and $V(y, n)$ be the open neighbourhoods of $x$ and $y$ in $U$ given by Proposition 4.33. Then there is some

$$ (x_n, y_n) \in Y(K, C) \cap (V(x, n) \times V(y, Cn)) . $$

According to Proposition 4.33, by taking $K$ and $C$ large enough, it can assumed that all $\phi_n = \phi_{x,x_n,n}$ and $\psi_n = \phi_{y,y_n,Cn}$ are non-expanding bi-lipschitz maps with bi-lipschitz distortion $C$, and the sets $B(x_n, n/C) \cap \text{im} \phi_n$ and $B(y_n, n) \cap \text{im} \psi_n$ are $K$-nets in $B(x_n, n/C)$ and $B(y_n, n)$, respectively. In particular, the restriction

$$ \phi_n : \phi_n^{-1}(B(x, n/C)) \to B(x_n, n/C) \cap \text{im} \phi_n $$

is a coarse quasi-isometry between $\phi_n^{-1}(B(x_n, n/C))$ and $B(x_n, n/C)$ with coarse distortion $(K, C)$.

On the other hand, for each $n$, there is a coarse quasi-isometry $f_n : A_n \to B_n$ between $\mathcal{G}(x_n)$ and $\mathcal{G}(y_n)$ with coarse distortion $(K, C)$, and so that $x_n \in A_n$, $y_n \in B_n$ and $f_n(x_n) = y_n$. Each set $B(x_n, n/C) \cap A_n$ is a $2K$-net in $B(x_n, n/C)$; this holds for $n > CK$ by Lemma 3.8, and for $n \le CK$ because $x_n \in B(x_n, n/C) \cap A_n$. Also, note that

$$ f_n(B(x_n, n/C) \cap A_n) \subset B(y_n, n), $$

and thus each $f_n(B(x_n, n/C) \cap A_n)$ is a $2K$-net in

$$ P_n = B(y_n, n) \cap \text{Pen}(f_n(B(x_n, n/C) \cap A_n), 2K), $$

yielding that each restriction

$$ f_n : B(x_n, n/C) \cap A_n \to f_n(B(x_n, n/C) \cap A_n) $$

is a coarse quasi-isometry between $B(x_n, n/C)$ and $P_n$ with coarse distortion $(2K, C)$. Moreover, because each $B(y_n, n) \cap \text{im} \psi_n$ is a $K$-net in $B(y_n, n)$, each $P_n \cap \text{im} \psi_n$ is a $2K$-net in $P_n$ by Lemma 3.8. So the restriction

$$ \psi_n^{-1} : P_n \cap \text{im} \psi_n \to \psi_n^{-1}(P_n) $$

is a coarse quasi-isometry between $P_n$ and $\psi_n^{-1}(P_n)$ with coarse distortion $(2K, C)$. It follows from Proposition 3.3 that, for some integers $K' \ge K$ and $C' \ge C$, there is a coarse quasi-isometry $g_n$ between $\phi_n^{-1}(B(x_n, n/C))$ and $\psi_n^{-1}(P_n)$ with coarse distortion $(K', C')$, and such that $x \in \text{dom } g_n$, $y \in \text{im } g_n$ and $g_n(x) = y$; this $g_n$ represents the “composite” of the above three coarse quasi-isometries.

Observe that

$$ B(x, n/C) \subset \phi_n^{-1}(B(x_n, n/C)) $$
because $\phi_n$ is non-expanding and $\phi_n(x) = x_n$. So the sequence of finite sets $\phi_n^{-1}(B(x_n, n/C))$ is exhausting in $\mathcal{S}(x)$. On the other hand, we have

$$B(y_n, n/C^2) \cap B_n \subset f_n(B(x_n, n/C) \cap A_n)$$

since $f_n: A_n \to B_n$ is a bi-lipschitz bijection with bi-lipschitz distortion $C$ such that $x_n \in A_n$, $y_n \in B_n$ and $f_n(x_n) = y_n$. Furthermore $B(y_n, n/C^2) \cap B_n$ is a $2K$-net in $B(y_n, n/C^2)$, which holds for $n > C^2K$ by Lemma 3.8, and for $n \le C^2K$ because $y_n \in B(y_n, n/C^2) \cap B_n$. Therefore

$$P_n \supset B(y_n, n) \cap \text{Pen}(B(y_n, n/C^2) \cap B_n, 2K) \supset B(y_n, n/C^2),$$

yielding

$$B(y, n/C^2) \subset \psi_n^{-1}(B(y_n, n/C^2)) \subset \psi_n^{-1}(P_n)$$

since $\psi_n$ is non-expanding and $\psi_n(y) = y_n$. So the sequence of finite sets $\psi_n^{-1}(P_n)$ is exhausting in $\mathcal{S}(y)$.

Now, by applying Proposition 3.7 to the sequence of coarse quasi-isometries $g_n$, it follows that $(x, y) \in Y(K', C')$. $\square$

**Remark.** If the statement of Lemma 5.2 were restricted to the residual union of orbits which do not meet the boundaries in $U$ of the domains of maps in $E$, then it would be an easy consequence of Propositions 3.7 and 4.32.

Let $Y$ be the set of points $(x, y) \in U_0 \times U_0$ such that $\mathcal{S}(x)$ is coarsely quasi-isometric to $\mathcal{S}(y)$.

**Lemma 5.3.** The set $Y = \bigcup_{K,C} Y(K, C)$.

*Proof.* This equality follows from Proposition 3.6. $\square$

**Corollary 5.4.** The set $Y$ is equal to the union of the closures in $U_0 \times U_0$ of the sets $Y(K, C)$; in particular, $Y$ is an $F_\delta$-subset of $U_0 \times U_0$.

*Proof.* This is elementary by Lemmas 5.2 and 5.3. $\square$

Now Theorem 5.1 follows from Corollary 5.5 since $U_0$ is residual in $U$ (Theorem 4.10).

**Corollary 5.5.** Either some set $Y(K, C)$ has non-empty interior in $U_0 \times U_0$, or else $Y$ is meager subset of $U_0 \times U_0$.

*Proof.* If all sets $Y(K, C)$ have empty interior in $U_0 \times U_0$, their closures have empty interior as well by Lemma 5.2, and thus $Y$ is meager in $U_0 \times U_0$ by Corollary 5.5. $\square$

**Theorem 5.6.** With the above notation, suppose that $\mathcal{H}$ is transitive. Then either all dense $\mathcal{S}$-orbits in $U_0$ are uniformly coarsely quasi-isometric to each other, or else there exists a residual $\mathcal{S}$-saturated subset $A \subset U_0$ such that, for any $\mathcal{S}$-orbit $\mathcal{O} \subset A$, the union of orbits that are coarsely quasi-isometric to $\mathcal{O}$ is a meager set in $U$; in particular, there are uncountably many coarsely quasi-isometry types of $\mathcal{S}$-orbits in the second case.
*Proof.* Suppose that some set $Y(K, C)$ has non-empty interior in $U_0 \times U_0$. Then all dense $\mathcal{G} \times \mathcal{G}$-orbits in $U_0 \times U_0$ meet $Y(K, C)$, and thus all dense $\mathcal{G}$-orbits in $U_0$ are uniformly coarsely quasi-isometric to each other.

If the interior of every set $Y(K, C)$ is empty, then $Y$ is meager in $U_0 \times U_0$ by Corollary 5.5. It follows from Theorem 4.2 that there is a residual subset $A \subset X_0$ such that $Y_x = \{y \in U_0 | (x, y) \in Y\}$ is meager in $U$ for all $x \in A$. But each $Y_x$ is the union of orbits in $U_0$ which are coarsely quasi-isometric to $\mathcal{G}(x)$. Finally, $A$ can be assumed to be $\mathcal{G}$-saturated since $Y$ is $\mathcal{G} \times \mathcal{G}$-saturated. $\square$

The next step consists in showing that when some $Y(C, K)$ has non-empty interior, the $\mathcal{G}$-orbits must have a very strong geometric property which is defined as follows.

**Definition 5.7.** A family $\mathcal{F}$ of coarsely quasi-isometric transformations of a metric space $M$ is called *transitive* when, for any $x, y \in M$, there is some $f \in \mathcal{F}$ such that $x \in \text{dom } f$, $y \in \text{im } f$ and $f(x) = y$. A metric space $M$ said to be *coarsely quasi-symmetric* if there is a transitive uniform family of coarse quasi-isometric transformations of $M$.

**Lemma 5.8.** Let $\mathcal{F}$ be a uniform family of coarse quasi-isometries of a metric space $M$. Suppose that there is some $R > 0$ such that, for every $x, y \in M$, there is some $f \in \mathcal{F}$ and some $z \in \text{dom } f$ such that

$$f(B_M(x, R) \cap \text{dom } f) \cap B_M(y, R) \neq \emptyset.$$

Then $M$ is coarsely quasi-symmetric.

*Proof.* This is a direct consequence of Proposition 3.6. $\square$

**Lemma 5.9.** The property of being coarsely quasi-symmetric is invariant by coarse quasi-isometries.

*Proof.* Let $M, M'$ be metric spaces with metrics $d, d'$, and let $h$ be a coarse quasi-isometry between $M$ and $M'$ of coarse distortion $(K, C)$ for some $K > 0$ and $C \ge 1$. Assume that $M$ is coarsely quasi-symmetric; thus there is a transitive uniform family $\mathcal{F}$ of coarse quasi-isometric transformations of $M$; we can suppose that $(K, C)$ also is a coarse distortion of every $f \in \mathcal{F}$. For each $f \in \mathcal{F}$, let $f'$ be the “coarse quasi-isometric composite” of $h^{-1}$, $f$ and $h$, defined by Proposition 3.3; more precisely, let $f'_0$ be the “coarse quasi-isometric composite” of $f$ and $h$, and let $f'$ be the “coarse quasi-isometric composite” of $h^{-1}$ and $f'_0$. According to Proposition 3.3, each $f'_0$ has coarse distortion $(6KC, 5C^2)$, and morever $\text{dom } f'_0 \subset \text{dom } f$, $\text{im } f'_0 \subset \text{im } h$, and

$$(5.1) \qquad d(f(x), h^{-1} \circ f'_0(x)) \le 2K$$

for every $x \in \text{dom } f'_0$. According to Proposition 3.3 again, each $f'$ has coarse distortion $(180KC^3, 125C^4)$, and morever $\text{dom } f' \subset \text{dom } h^{-1} = \text{im } h$, $\text{im } f' \subset \text{im } h$, and

$$(5.2) \qquad d(h^{-1}(x'), f_0'^{-1} \circ f'(x')) \le 12KC$$
for every $x' \in \text{dom } f'$; in particular, all maps $f'$ form a uniform family $\mathcal{F}'$ of coarsely quasi-isometric transformations of $M'$.

Now take arbitrary points $x', y' \in M'$. There are some $x, y \in \text{dom } h$ such that $d'(h(x), x') \le K$ and $d'(h(y), y') \le K$. Then there is some $f \in \mathcal{F}$ with $x \in \text{dom } f$, $y \in \text{im } f$ and $f(x) = y$. On the other hand, there is some $z' \in \text{dom } f'$ such that $d'(z', x') \le 180KC^3$, and let $z = h^{-1}(z')$. Hence

$$
\begin{align*}
d'(y', f'(z')) &\le d'(y', h(y)) + d'(h(y), f'(z')) \\
&\le K + C d(y, h^{-1} \circ f'(z')) \\
&\le K + C d(y, f(z)) + C d(f(z), h^{-1} \circ f'(z')) \\
&\le K + C d(f(x), f(z)) + C d(f(z), h^{-1} \circ f'_0(z)) + C d(h^{-1} \circ f'_0(z), h^{-1} \circ f'(z')) \\
&\le K + C^2 d(x, z) + 2KC + C^2 d(f'_0(z), f'(z')) \\
&\le K + 2KC + C^3 d(h(x), h(z)) + C^2 d(f'_0(z'), f'(z')) \\
&= K + 2KC + C^3 d'(h(x), z') + C^2 d(f'_0 \circ h^{-1}(z'), f'(z')) \\
&\le K + 2KC + C^3 d'(h(x), x') + C^3 d'(x', z') + 5C^4 d(h^{-1}(z'), f'_0^{-1} \circ f'(z')) \\
&\le K + 2KC + C^3 K + 180KC^6 + 60KC^5,
\end{align*}
$$

where (5.1) and (5.2) have been used. Therefore $\mathcal{F}'$ satisfies the hypotheses of Lemma 5.8, and thus $M'$ is coarsely quasi-symmetric. $\square$

**Theorem 5.10.** With the above notation, the following properties hold:

* Suppose that $\mathcal{H}$ is transitive. Then all $\mathcal{G}$-orbits in $U_0$ are uniformly coarsely quasi-isometric to each other if there is a coarsely quasi-symmetric dense $\mathcal{G}$-orbit in $U_0$.

* Suppose that $\mathcal{H}$ is minimal. Then all $\mathcal{G}$-orbits in $U_0$ are uniformly coarsely quasi-isometric to each other if and only if there is a coarsely quasi-symmetric dense $\mathcal{G}$-orbit in $U_0$.

*Proof.* Suppose that $\mathcal{H}$ is transitive and that there is a coarsely quasi-symmetric $\mathcal{G}$-orbit $\mathcal{O}$ in $U_0$. Then $\mathcal{O} \times \mathcal{O} \subset Y(K, C)$ for some integers $K, C \ge 1$, yielding $U_0 \times U_0 = Y(K', C')$ for some integers $K' \ge K$ and $C' \ge C$ by Lemma 5.2.

Now assume that $\mathcal{H}$ is minimal and that all $\mathcal{G}$-orbits in $U_0$ are uniformly coarsely quasi-isometric to each other. This means that some set $Y(K, C)$ has non-empty interior in $U_0 \times U_0$ by Lemma 5.3; i.e., there are some non-empty open subsets $V, W$ of $U_0$ so that $V \times W \subset Y(K, C)$. But, by Lemma 3.23 and since $\mathcal{G}$ is minimal, the intersections $\mathcal{O} \cap V$ and $\mathcal{O} \cap W$ are nets in $\mathcal{O}$ for any $\mathcal{G}$-orbit $\mathcal{O}$ in $U_0$. So any $\mathcal{G}$-orbit in $U_0$ is coarsely quasi-symmetric by Lemma 5.8. $\square$

Theorems 5.1, 5.6 and 5.10 have the following direct consequences for foliated spaces.

**Theorem 5.11.** Let $(X, \mathcal{F})$ be a compact foliated space, and let $X_0$ be the union of leaves without holonomy. Then the equivalence relation "$x \sim y$ if and only if the leaves trough $x$ and
y are coarsely quasi-isometric” has a borel relation set in $X_0 \times X_0$, and has a baire relation set in $X \times X$; in particular, it has borel equivalence classes in $X_0$, and baire equivalence classes in X.

**Theorem 5.12.** With the notation and conditions of Theorem 5.11, if $\mathcal{F}$ is transitive, then either all leaves without holonomy are uniformly coarsely quasi-isometric to each other, or else there exists a residual saturated subset $A \subset X_0$ such that, for any leaf $L \subset A$, the union of leaves that are coarsely quasi-isometric to $L$ is a meager set; in particular, $\mathcal{F}$ has uncountably many coarsely quasi-isometric types of leaves without holonomy in the second case.

**Theorem 5.13.** With the notation and conditions of Theorem 5.11, the following properties hold.

* Suppose that $\mathcal{F}$ is transitive. Then all leaves without holonomy are uniformly coarsely quasi-isometric to each other if there is a coarsely quasi-symmetric dense leaf without holonomy.

* Suppose that $\mathcal{F}$ is minimal. Then all leaves without holonomy are uniformly coarsely quasi-isometric to each other if and only if there is a coarsely quasi-symmetric dense leaf without holonomy.

**Remark.** *** *** *** version for orbits with finite groups of germs *** *** *** and for leaves with finite holonomy groups *** *** ***

******** measure theoretic versions **********

## 5.2 Ends of coarsely quasi-symmetric leaves

*** *** ***

**Theorem 5.14.** A connected coarsely quasi-symmetric metric space has one end, two ends or a cantor space of ends.

*Proof.***

☐

**Theorem 5.15.** A connected coarsely quasi-symmetric metric space with two ends is coarsely quasi-isometric to $\mathbb{Z}$.

*Proof.***

☐

**Theorem 5.16.** A connected coarsely quasi-symmetric metric space with a cantor space of ends is coarsely quasi-isometric to the free group with two generators.

*Proof.***

☐

*** *** *** consequences for pseudogroups and foliated spaces *** ***

******** for example, for minimal compact foliated spaces, if there is a leaf without holonomy whose end space is not one point, two points or a cantor space, then there are uncountably many coarsely quasi-isometric types of leaves ********
## 5.3 Growth of leaves

One of the classic geometric invariants of the leaves of a foliated space is the growth type, which is defined as follows.

Recall that an equivalence relation on the set of functions $[0, \infty) \to [0, \infty)$ can be defined as follows: two such functions $u, v$ are equivalent when there are some numbers $a, b \ge 1$ and $c \ge 0$ such that

$$v(r/b)/a \le u(r) \le a v(br)$$

for all $r \ge c$. Each equivalence class is called a *growth type*. An order relation between growth types can be defined by using only one of the above inequalities. Observe that the non-comparability of growth types represented by functions $u$ and $v$ means that there are sequences $r_n, r_n \uparrow \infty$ such that $u(r_n)/v(r_n) \to \infty$ and $v(s_n)/u(s_n) \to \infty$; so there is a sequence $R_n \uparrow \infty$ such that $u(R_n) = v(R_n)$ for all $n$. The growth types of a family of functions $[0, \infty) \to [0, \infty)$ is said to *uniformly equal* when the above equivalence relation holds for any pair of such functions with the same constants $a, b$ and $c$.

The *growth function* of a complete riemannian manifold $L$ is the function $v: L \times [0, \infty) \to [0, \infty)$ defined by $v(x, r) = \text{vol} B(x, r)$. The growth function provides a basic asymptotic invariant: the growth type of each mapping $r \mapsto v(x, r)$ is independent of $x$ and is called the *growth type* of $L$.

Other definitions of growth type can be similarly given for metric spaces whose bounded sets are finite, where vol refers to the number of points, and for simplicial metric spaces with a finite number of top dimensional simplices in every bounded set, where vol counts the number of top dimensional simplices.

For a metric space $(M, d)$ of coarse bounded geometry, the growth type can be defined as the growth type of any quasi-lattice $\Gamma$ of $M$; i.e., it is the growth type of the *growth function* $r \mapsto v_{\Gamma}(x, r) = \#(B(x, r) \cap \Gamma)$ for any $x \in \Gamma$. This definition can be proved to be independent of $\Gamma$ as follows. Let $\Gamma'$ be another quasi-lattice in $M$. So $\Gamma$ and $\Gamma'$ are $K$-nets in $M$ for some $K \ge 0$, and there are some $Q_r, Q'_r \ge 0$ for each $r \ge 0$ such that $\#(B(x, r) \cap \Gamma) \le Q_r$ and $\#(B(x, r) \cap \Gamma') \le Q'_r$ for all $x \in M$. Fix points $x \in \Gamma$ and $x' \in \Gamma'$, and let $\delta = d(x, x')$. Because $B(x, r) \subset B(x', r + \delta)$ and $\Gamma'$ is a $K$-net, it follows that

$$B(x, r) \cap \Gamma \subset \bigcup_{y' \in B(x', r+\delta+K) \cap \Gamma'} B(y', K) \cap \Gamma',$$

yielding

$$v_{\Gamma}(x, r) \le Q'_{K} v_{\Gamma'}(x', r + \delta + K) \le Q'_{K} v_{\Gamma'}(x', (1 + \delta + K)r)$$

for all $r \ge 1$. Hence the growth type of the mapping $r \mapsto v_{\Gamma}(x, r)$ is less or equal than the growth type of the mapping $r \mapsto v_{\Gamma'}(x, r)$, and the reverse inequality follows similarly.

Observe that this definition of growth type coincides with the above definition in the case of a complete riemannian manifold of bounded geometry, as well as in the case of discrete or simplicial metric spaces with a uniform upper bound on the number of points or top dimensional simplices in all balls of each given radius.
**Lemma 5.17.** *Two coarsely quasi-isometric metric spaces of coarse bounded geometry have the same growth type.*

*Proof.* Let $\phi: A \to A'$ be a coarse quasi-isometry between metric spaces $M$ and $M'$ of coarse bounded geometry. Then $A$ is of coarse bounded geometry too, and thus it has some lattice $\Gamma$, which is also a lattice in $M$ because $A$ is a net. Since $\phi$ is a bi-lipschitz bijection, it easily follows that $\Gamma$ and $\phi(\Gamma)$ have the same growth type, and that $\phi(\Gamma)$ is a lattice in $A'$, and thus in $M'$ too because $A'$ is a net. $\square$

The growth types of a family of metric spaces of coarse bounded geometry is said to uniformly equal when there is a lattice in each metric space and a point in each lattice so that the corresponding growth functions define the same growth type uniformly. A similar definition can be given for families of riemannian manifolds, and discrete or simplicial metric spaces.

For a compact foliated space $(X, \mathcal{F})$, the growth type of the leaves can be defined by using any good metric on $R_{\mathcal{F}}$, and it coincides with the growth type of the orbits of any appropriate representative of the corresponding orbits of the holonomy pseudogroup by Theorem ?? and Lemma 5.17. This section analyzes the behavior of the growth function of the leaves as a function of $X$. As in the above section, the pseudogroup version of this problem is considered first. Then, with the notation of the above section, the growth type of the $\mathcal{G}$-orbit of each $x \in U$ is the growth type of the mapping $r \mapsto v_E(x, r) = \#B_E(x, r)$.

**Theorem 5.18.** *With the notation and conditions of Theorem 5.1, the equivalence relation “$x \sim y$ if and only if the orbits $\mathcal{G}(x)$ and $\mathcal{G}(y)$ have the same growth type” has a borel relation set in $U_0 \times U_0$, and has a baire relation set in $U \times U$; in particular, it has borel equivalence classes in $U_0$, and baire equivalence classes in $U$.*

*Proof.* For $a, b \ge 1$ and $c \ge 0$, let

$$Y(a, b, c) = \{(x, y) \in U_0 \times U_0 \mid v_E(x, r) \le a v_E(y, br) \quad \forall r \ge c\}.$$

**Claim 10.** *For each $a, b \ge 1$ and $c \ge 0$, there are some $a' \ge a$ and $b' \ge b$ such that $Y(a', b', c)$ contains the closure of $Y(a, b, c)$ in $U_0 \times U_0$.*

Consider the notation of Proposition 4.33 to prove this claim. Let $(x, y)$ be a point in the closure of $Y(a, b, c)$ in $U_0 \times U_0$. For any $r \ge c$, if

$$(x', y') \in Y(a, b, c) \cap (V(x, r) \times V(y, Cbr)),$$

then

$$v_E(x, r) \le v_E(x', r) \le a v_E(y', br) \le a(\#E)^K v_E(y, Cbr)$$

since $\phi_{x,x',r}$ is injective and $B_E(y', br) \cap \text{im} \phi_{y,y',Cbr}$ is a $K$-net in $B_E(y', br)$ by Proposition 4.33, and because $\#B_E(z, K+1) \le (\#E)^K$. Hence $(x, y) \in Y(a(\#E)^K, Cb, c)$.
By Claim 10, the union

$$Y = \bigcup_{a,b \ge 1, c \ge 0} Y(a, b, c) = \bigcup_{a,b,c=1}^{\infty} Y(a, b, c)$$

is an $F_\delta$-subset of $U_0 \times U_0$, and thus so is $Z = Y \cap \sigma(Y)$, where $\sigma: U \times U \to U \times U$ is the homeomorphism given by $\sigma(x, y) = (y, x)$. But $Z$ is the relation set in $U_0 \times U_0$ of the statement, and the result follows because $U_0$ is residual in $U$ (Theorem 4.22). □

**Theorem 5.19.** With the notations and conditions of Theorem 5.18, if $\mathcal{G}$ is transitive, then either all dense $\mathcal{G}$-orbits in $U_0$ have the same growth type uniformly, or else there exists a residual $\mathcal{G}$-saturated subset $A \subset U_0$ such that, for any $x \in A$, the union of $\mathcal{G}$-orbits whose growth type is comparable with the growth type of $\mathcal{G}(x)$ is a meager set; in particular, there are uncountably many growth types of $\mathcal{G}$-orbits in $U_0$ in this second case.

*Proof.* With the notation of the proof of Theorem 5.18, suppose that some set $Y(a, b, c)$ has non-empty interior in $U_0 \times U_0$. Then all dense orbits of $\mathcal{G} \times \mathcal{G}$ in $U_0 \times U_0$ meet $Y(a, b, c)$. It follows that all dense orbits of $\mathcal{G}$ in $U_0$ have the same growth type uniformly.

On the other hand, if all sets $Y(a, b, c)$ have empty interior in $U_0 \times U_0$, then $Y$ is a meager subset of $U_0 \times U_0$ by Claim 10, and thus it is meager in $U \times U$ too. So $Y' = Y \cup \sigma(Y)$ is meager in $U \times U$ as well. It follows from Theorem 4.2 that there is a residual subset $A \subset U_0$ such that $Y'_x = \{y \in U_0 \mid (x, y) \in Y'\}$ is meager for all $x \in A$. But each $Y'_x$ is the union of $\mathcal{G}$-orbits in $U_0$ whose growth type is comparable with the growth type of the $\mathcal{G}(x)$. Obviously, it can be assumed that $A$ is saturated. □

**Definition 5.20.** A metric space $M$ of coarse bounded geometry is said to be *growth symmetric* if there is some lattices $\Gamma, \Gamma'$ in $M$, and there are some constants $a, b \ge 1$ and $c \ge 0$ such that $v_\Gamma(x, r) \le a v_{\Gamma'}(y, br)$ for all $r \ge c$, $x \in \Gamma$ and $y \in \Gamma'$.

**Remark.** This definition is easily proved to be independent of the choices of $\Gamma$ and $\Gamma'$, and invariant by coarse quasi-isometries.

Similar definitions of growth symmetry can be given for riemannian manifolds, metric spaces whose bounded sets are finite, and simplicial metric spaces whose bounded sets have a finite number of top dimensional simplices. Moreover, in the case of complete riemannian manifolds of bounded geometry, both definitions of growth symmetry are equivalent, and the same holds for appropriate discrete and simplicial metric spaces.

**Theorem 5.21.** With the notation and conditions of Theorem 5.18, the following properties hold:

* Suppose that $\mathcal{G}$ is transitive. Then all $\mathcal{G}$-orbits in $U_0$ have the same growth type uniformly if there is a growth symmetric dense $\mathcal{G}$-orbit in $U_0$.

* Suppose that $\mathcal{G}$ is minimal. Then all $\mathcal{G}$-orbits in $U_0$ have the same growth type uniformly if and only if there is a growth symmetric $\mathcal{G}$-orbit in $U_0$.
*Proof.* Suppose that $\mathcal{G}$ is transitive and that $\mathcal{G}(x)$ is dense and growth symmetric for some $x \in U_0$. Then $\mathcal{G}(x) \times \mathcal{G}(x) \subset Y(a, b, c)$ for some $a, b \ge 1$ and $c \ge 0$, yielding $U_0 \times U_0 = Y(a', b', c)$ for some $a, b \ge 1$ by Claim 10.

Now assume that $\mathcal{G}$ is minimal and that all $\mathcal{G}$-orbits in $U_0$ have the same growth type uniformly. This means that some set $Y(a, b, c)$ has non-empty interior in $U_0 \times U_0$ according to the proof of Theorem 5.19; i.e., there are some non-empty open subsets $V, W$ of $U_0$ such that $V \times W \subset Y(a, b, c)$. But, since $\mathcal{G}$ is minimal, the intersections $\mathcal{G}(x) \cap V$ and $\mathcal{G}(x) \cap W$ are nets in $\mathcal{G}(x)$ for any $x \in U_0$ by Lemma 3.23. So any $\mathcal{G}$-orbit in $U_0$ is growth symmetric. $\square$

**Theorem 5.22.** With the notations and conditions of Theorem 5.18, suppose that $\mathcal{G}$ is transitive. Then any of the following sets is either meager or residual in $X$:

* The union of $\mathcal{G}$-orbits in $U_0$ that have polynomial growth;

* the union of $\mathcal{G}$-orbits in $U_0$ that have exponential growth;

* The union of $\mathcal{G}$-orbits in $U_0$ that have quasi-polynomial growth; and

* the union of $\mathcal{G}$-orbits in $U_0$ that have quasi-exponential growth.

Moreover any if any of the first two sets is residual in $X$, then it is equal to the union of dense $\mathcal{G}$-orbits in $U_0$.

*Proof.* For $a, b > 0$, let $Y(a, b)$, $Z(a, b)$, $Y'(a, b)$ and $Z'(a, b)$ be the sets of points $x \in U_0$ that respectively satisfy the following conditions

$$ \sup_{r \ge b} \frac{\log v_E(x, r)}{\log r} \le a, \quad \inf_{r \ge b} \frac{\log v_E(x, r)}{r} \ge a, $$

$$ \inf_{r \ge b} \frac{\log v_E(x, r)}{\log r} < a, \quad \sup_{r \ge b} \frac{\log v_E(x, r)}{r} > a. $$

**Claim 11.** The following properties hold for each $a, b > 0$:

* $Y(a, b)$ is closed in $U_0$;

* there are some $b' \ge b$ such that $Z(a, b')$ contains the closure of $Z(a, b)$ in $U_0$;

* there are some $a' \ge a$ such that $Y'(a, b)$ is contained in the interior of $Y'(a', b)$ in $U_0$;

* $Z'(a, b)$ is open in $U_0$.

Consider the notation of Proposition 4.33 to prove this claim.

Let $x$ be a point in the closure of $Y(a, b)$ in $U_0$. For any $r \ge b$, if $y \in Y(a, b) \cap V(x, r)$, then

$$ \frac{\log v_E(x, r)}{\log r} \le \frac{\log v_E(y, r)}{\log r} \le a $$

because $\phi_{x,y,r}$ is injective by Proposition 4.33. So $x \in Y(a, b)$.
Now, let $x$ be a point in the closure of $Z(a, b)$ in $U_0$. For any $r \ge Cb$, if $y \in Z(a, b) \cap V(x, r)$, then

$$ \frac{\log v_E(x, r)}{r} \ge (\#E)^K \frac{\log v_E(y, r/C)}{\log(r/C)} \ge (\#E)^K a \ge a $$

since $\phi_{x,y,r}$ is injective and $B_E(y, r/C) \cap \text{im} \phi_{x,y,r}$ is a $K$-net in $B_E(y, r)$ by Proposition 4.33. Thus $x \in Z(a, Cb)$.

For any $x \in Y(a, b)$, there is some $r \ge b$ such that $\log v_E(x, r)/\log r < a$. If $y \in V(x, r)$, then

$$ \frac{\log v_E(y, Cr)}{\log Cr} \le \frac{\log v_E(y, Cr)}{\log r} \le (\#E)^K \frac{\log v_E(x, r)}{\log r} < (\#E)^K a $$

because $B_E(y, r) \cap \text{im} \phi_{x,y,Cr}$ is a $K$-net in $B_E(y, r)$ by Proposition 4.33. Hence $V(x, r) \cap U_0 \subset Y'((\#E)^K a, b)$, and thus $x$ lies in the interior of $Y'((\#E)^K a, b)$ in $U_0$.

Finally, for any $x \in Z'(a, b)$, there is some $r \ge b$ such that $\log v_E(x, r)/r > a$. Hence $\log v_E(y, r)/r > a$ for any $y \in V(x, r)$ since $\phi_{x,y,r}$ is injective, yielding $V(x, r) \cap U_0 \subset Z'(a, b)$. Therefore $Z'(a, b)$ is open in $U_0$, which finishes the proof of Claim 11.

Let

$$ Y = \bigcup_{a,b} Y(a,b), \quad Z = \bigcup_{a,b} Z(a,b), \quad Y' = \bigcup_{a} \bigcap_{b} Y'(a,b), \quad Z' = \bigcup_{a} \bigcap_{b} Z'(a,b). $$

Since these expressions are the same when $a, b$ run on positive rational numbers, it follows from Claim 11 that $Y, Z, Y'$ and $Z'$ are borel subsets of $U_0$, and thus baire subsets of $U$. Moreover, it is clear that $Y, Z, Y'$ and $Z'$ are the sets of points $x \in U_0$ that respectively satisfy the following conditions

$$ \limsup_{r \to \infty} \frac{\log v_E(x, r)}{\log r} < \infty, \quad \liminf_{r \to \infty} \frac{\log v_E(x, r)}{r} > 0, $$

$$ \liminf_{r \to \infty} \frac{\log v_E((x, r))}{\log r} < \infty, \quad \limsup_{r \to \infty} \frac{\log v_E(x, r)}{r} > 0. $$

Thus $Y, Z, Y', Z'$ are the unions of $\mathcal{G}$-orbits in $U_0$ that respectively have polynomial growth, exponential growth, quasi-polynomial growth, and quasi-exponential growth $Y, Z, Y', Z'$ are saturated. Hence any of $Y, Z, Y', Z'$ is either residual or meager in $U$ by Theorem 4.9.

If $Y$ (respectively, $Z$) is residual in $U$, then some $Y(a, b)$ (respectively, $Z(a, b)$) has non-empty interior in $U_0$ by Claim 11; thus it meets all dense $\mathcal{G}$-orbits in $U_0$, and the last part of the result follows. $\square$

**Theorem 5.23.** With the notations and conditions of Theorem 5.18, suppose that $\mathcal{G}$ is transitive. Then the functions $U \to [0, \infty]$, defined by

$$ x \mapsto \limsup_{r \to \infty} \frac{\log v_E(x, r)}{\log r}, \quad x \mapsto \liminf_{r \to \infty} \frac{\log v_E(x, r)}{r}, $$

$$ x \mapsto \liminf_{r \to \infty} \frac{\log v_E(x, r)}{\log r}, \quad x \mapsto \limsup_{r \to \infty} \frac{\log v_E(x, r)}{r}, $$

are constant on some residual subset of $U$.
Proof. If $x, y$ belong to the same $\mathcal{G}$-orbit, and $R = d_E(x, y)$, then

$$v_E(x, r) \le v_E(y, r + R) \le v_E(x, r + 2R),$$

yielding that the functions $X \to [0, \infty]$ of the statement are constant on the $\mathcal{G}$-orbits. In particular, for each $a \in [0, \infty]$, the sets

$$Y(a) = \left\{ x \in U_0 \mid \limsup_{r \to \infty} \frac{\log v_E(x, r)}{\log r} \le a \right\}, \quad Z(a) = \left\{ x \in U_0 \mid \liminf_{r \to \infty} \frac{\log v_E(x, r)}{r} \ge a \right\},$$

$$Y'(a) = \left\{ x \in U_0 \mid \liminf_{r \to \infty} \frac{\log v_E(x, r)}{\log r} \le a \right\}, \quad Z'(a) = \left\{ x \in U_0 \mid \limsup_{r \to \infty} \frac{\log v_E(x, r)}{r} \ge a \right\}$$

are saturated. Moreover, with the notation of the proof of Theorem 5.22, we have

$$Y(a) = \bigcap_{c>a} \bigcup_{b} Y(c, b), \quad Z(a) = \bigcap_{c<a} \bigcup_{b} Z(c, b), \\ Y'(a) = \bigcap_{c>a} \bigcap_{b} Y'(c, b), \quad Z'(a) = \bigcap_{c<a} \bigcap_{b} Z'(c, b),$$

which are borel subsets of $U_0$ by Claim 11 because the same equalities hold with $c, b$ running on the positive rational numbers. Therefore each of the sets $Y(a), Z(a), Y'(a)$ and $Z'(a)$ is either residual or meager in $U$ by Theorem 4.9. We obviously have

$$Y(\infty) = Z(0) = Y'(\infty) = Z'(0) = U_0.$$

Then let

$$a_0 = \inf\{a \in [0, \infty] \mid Y(a) \text{ is residual in } U\},$$

$$a_1 = \sup\{a \in [0, \infty] \mid Z(a) \text{ is residual in } U\},$$

$$a'_0 = \inf\{a \in [0, \infty] \mid Y'(a) \text{ is residual in } U\},$$

$$a'_1 = \sup\{a \in [0, \infty] \mid Z'(a) \text{ is residual in } U\}.$$

Because the growth type of an infinite connected graph of finite type is at least linear and at most exponential, the sets $Y(a), Y'(a)$ are unions of finite orbits for $0 \le a < 1$, and $Z(\infty) = Z'(\infty) = \emptyset$. Moreover

$$Y(a_0) = \bigcap_{a_0 < a < \infty} Y(a) \quad \text{if } a_0 < \infty,$$

$$Z(a_1) = \bigcup_{0 < a < a_1} Z(a) \quad \text{if } a_1 > 0,$$

$$Y'(a'_0) = \bigcap_{a_0 < a < \infty} Y'(a) \quad \text{if } a'_0 < \infty,$$

$$Z'(a'_1) = \bigcup_{0 < a < a'_1} Z'(a) \quad \text{if } a'_1 > 0,$$
where $a$ can be taken to be a rational number. Thus $Y(a_0)$, $Z(a_1)$, $Y'(a'_0)$, $Z'(a'_1)$ are residual in $U$. On the other hand, the unions

$$ \bigcup_{0 \le a < a_0} Y(a), \quad \bigcup_{a_1 < a < \infty} Z(a), \quad \bigcup_{0 \le a < a'_0} Y'(a), \quad \bigcup_{a'_1 < a < \infty} Z'(a), $$

do not change if $a$ is assumed to be rational, and thus they are meager sets. It follows that

$$ \limsup_{r \to \infty} \frac{\log v(x, r)}{\log r} = a_0, \quad \liminf_{r \to \infty} \frac{\log v(x, r)}{r} = a_1, $$

$$ \liminf_{r \to \infty} \frac{\log v(x, r)}{\log r} = a'_0, \quad \limsup_{r \to \infty} \frac{\log v(x, r)}{r} = a'_1, $$

for all $x$ in a saturated residual subset of $U_0$.

<figure><img src="image_9.png" /></figure>

Theorems 5.18–5.22 have the following direct consequences for foliated spaces.

**Theorem 5.24.** Let $(X, \mathcal{F})$ be a compact foliated space, and let $X_0$ be the union of leaves without holonomy. Then the equivalence relation "$x \sim y$ if and only if the leaves trough $x$ and $y$ have the same growth type" has a borel relation set in $X_0 \times X_0$, and has a baire relation set in $X \times X$; in particular, it has borel equivalence classes in $X_0$, and baire equivalence classes in $X$.

**Theorem 5.25.** With the notations and conditions of Theorem 5.24, if $\mathcal{F}$ is transitive, then either all dense leaves without holonomy have the same growth type uniformly, or else there exists a residual saturated subset $A \subset X_0$ such that, for any leaf $L \subset A$, the union of leaves whose growth type is comparable with the growth type of $L$ is a meager set; in particular, there are uncountably many growth types of leaves without holonomy in this second case.

**Theorem 5.26.** With the notation and conditions of Theorem 5.24, the following properties hold:

* Suppose that $\mathcal{F}$ is transitive. Then all leaves without holonomy have the same growth type uniformly if there is a growth symmetric dense leaf without holonomy.

* Suppose that $\mathcal{F}$ is minimal. Then all leaves without holonomy have the same growth type uniformly if and only if there is a growth symmetric dense leaf without holonomy.

**Theorem 5.27.** With the notations and conditions of Theorem 5.24, suppose that $\mathcal{F}$ is transitive. Then any of the following sets is either meager or residual in $X$:

* The union of leaves without holonomy that have polynomial growth;

* the union of leaves without holonomy that have exponential growth;

* The union of leaves without holonomy that have quasi-polynomial growth; and

* the union of leaves without holonomy that have quasi-exponential growth.
Moreover if any of the first two sets is residual in X, then it is equal to the union of dense leaves without holonomy.

For any $C^1$ foliated space $(X, \mathcal{F})$ and any riemannian metric on the leaves, the following result can be proved like Theorem 5.23 by using $v(x, r) = \text{vol } B_{\mathcal{F}}(x, r)$ instead of $v_E(x, r)$, and by applying Theorem 4.21 and Proposition 2.23 instead of Theorem 4.9 and Proposition 4.33.

**Theorem 5.28.** With the notations and conditions of Theorem 5.24, suppose that $\mathcal{F}$ is transitive and $C^1$. Then, for any choice of a riemannian metric on the leaves of $\mathcal{F}$, continuous on $X$, the functions $X \to [0, \infty]$, defined by

$$x \mapsto \limsup_{r \to \infty} \frac{\log v(x, r)}{\log r},$$

$$x \mapsto \liminf_{r \to \infty} \frac{\log v(x, r)}{r},$$

$$x \mapsto \liminf_{r \to \infty} \frac{\log v(x, r)}{\log r},$$

$$x \mapsto \limsup_{r \to \infty} \frac{\log v(x, r)}{r},$$

are constant on some residual subset of X.

*** measure theoretic versions ***

## 5.4 Amenable leaves

Amenability of the leaves of compact foliated spaces is studied in this section. It is proved that, either no dense leaf without holonomy is amenable, or else all dense leaves are amenable in a uniform way; in this case, if the foliated space is transitive, then the leaves are shown to have certain uniformly symmetric amenability.

Consider the terminology and notation of Section 3.5). Let $\Gamma_i$ be a lattice in each $M_i$ of a given family $\{M_i\}$ of metric spaces. The family of lattices $\{\Gamma_i\}$ is said to be a *uniform* if there is some $K \ge 0$ and some $Q_r > 0$ for each $r > 0$ such that every $\Gamma_i$ is a $K$-net in $M_i$, and $\#(B_{M_i}(x, r) \cap \Gamma) \le Q_r$ for all $x \in M_i$. The metric spaces $M_i$ are said to be *uniformly amenable* when they have a uniform family of lattices $\Gamma_i$ such that there is a *fölner sequence* $S_{i,n}$ in each $\Gamma_i$ so that the sequences $\#\partial_r^{\Gamma_i} S_{i,n} / \#S_{i,n}$ have the same order of convergence to zero as $n \to \infty$ for all $r > 0$.

A *uniform fölner* condition can be similarly defined for families of riemannian manifolds, metric spaces whose bounded sets are finite, and simplicial metric spaces whose bounded sets have a finite number of top dimensional simplices. Then a family of connected graphs of *uniform finite type* (there is an upper bound on the number of edges that meet at every vertex of all of them) is uniformly amenable if and only if it is uniformly *fölner*. Also, a family of complete riemannian manifolds of *uniformly bounded geometry* (there a common lower bound for the injectivity radius and a common upper bound for the pointwise norm of the covariant derivative of each order of the curvature) is uniformly amenable if and only if it is uniformly *fölner*.

A metric space $M$ is said to have *symmetric amenability* if it has some lattice $\Gamma$ and a family of *fölner* sequences $S_{\alpha,n}$ in $\Gamma$ so that the sequences $\#\partial_r^{\Gamma} S_{\alpha,n} / \#S_{\alpha,n}$ have the same
order of convergence to zero as $n \to \infty$ for all $r > 0$, and $\bigcup_{\alpha} S_{\alpha,n}$ is a net in $\Gamma$ (and thus in $M$) for every $n$. It is easy to check that this definition is independent of the choice of $\Gamma$, and invariant by coarse quasi-isometries.

A definition of *fölner symmetry* can be similarly given for riemannian manifolds, metric spaces whose bounded sets are finite, and simplicial metric spaces whose bounded sets have a finite number of top dimensional simplices. Moreover complete riemannian manifolds of bounded geometry and connected graphs of finite type have symmetric amenability if and only if they are *fölner symmetric*. Also, definitions of *uniformly symmetric amenability* and *uniformly *fölner symmetry* can be given for such families in the obvious way.

**Lemma 5.29.** Let $M$ be a connected graph of finite type. If $M$ is *fölner*, then there is some *fölner sequence* $S_n$ in $M$ such that the sequences

$$ \frac{\#\partial S_n}{\#S_n}, \quad \frac{\#(\mathrm{Pen}(S_n, r+1) \setminus \mathrm{Pen}(S_n, r))}{\#S_n} $$

have the same order of convergence to zero as $n \to \infty$ for any fixed $r > 0$.

*Proof.* Let $Q$ be an upper bound on the number of edges that meet at each vertex. Thus

$$ \# \mathrm{Pen}(S, r) \le Q^r \cdot \#S $$

for any finite $S \subset M$ and any $r \ge 0$. Then, since

$$ \mathrm{Pen}(S, r+1) \setminus \mathrm{Pen}(S, r) \subset \partial_{r+2} S \subset \mathrm{Pen}(\partial S, r+1), $$

it follows that

$$ (5.3) \qquad \# \mathrm{Pen}(S, r+1) \setminus \mathrm{Pen}(S, r) \le Q^{r+1} \cdot \#\partial S. $$

**Claim 12.** For any $S \subset M$ and $r \ge 1$, let

$$ S' = \mathrm{Pen}(S, r) \setminus \mathrm{Pen}(X \setminus \mathrm{Pen}(S, r), r). $$

Then

$$ \partial S' \subset \mathrm{Pen}(\mathrm{Pen}(S', r+1) \setminus \mathrm{Pen}(S', r), r+1). $$

Note that

$$ (5.4) \qquad d(u, S) > r \implies d(x, u) > r $$

for all $x \in S'$ and $u \in M$. Now fix any $x \in \partial S'$ to prove the above inclusion; i.e., $x \in S'$ and there is some $y \in M \setminus S'$ such that $d(x, y) = 1$, where $d$ denotes the graph metric on $M$. So, either $d(y, S) > r$, or else there is some $z \in M$ such that $d(z, S) > r$ and $d(y, z) \le r$. But $d(y, S) > r$ contradicts $d(x, y) = 1$ by (5.4); thus the existence of such a $z$ is guaranteed, and moreover

$$ (5.5) \qquad d(x, z) \le d(x, y) + d(y, z) \le 1 + r. $$
Suppose that $d(z, S') \le r$. Then there exists some $x' \in S'$ such that $d(x', z) \le r$, and $d(z, S) > r$ yields the contradiction $d(x', z) > r$ by (5.4). Therefore $d(z, S') > r$, yielding

$$d(z, S') = d(z, x) = r + 1,$$

thus

$$z \in \mathrm{Pen}(S', r+1) \setminus \mathrm{Pen}(S', r),$$

and

$$x \in \mathrm{Pen}(\mathrm{Pen}(S', r+1) \setminus \mathrm{Pen}(S', r), r+1),$$

showing Claim 12.

According to Claim 12, for any finite $S \subset M$ and $r \ge 1$, the corresponding set $S'$ is finite and satisfies

$$(5.6) \qquad \#\partial S' \le Q^{r+1} \cdot \#(\mathrm{Pen}(S', r+1) \setminus \mathrm{Pen}(S', r)).$$

By (5.3) and (5.6), for any fölner sequence $S_n$ in $M$, the corresponding sequence $S'_n$ satisfies the statement of the lemma. $\square$

**Theorem 5.30.** With the notation and conditions of Theorem 5.1, if some $\mathcal{G}$-orbit in $U_0$ is fölner, then all dense $\mathcal{G}$-orbits are uniformly fölner. If $\mathcal{G}$ is minimal and some $\mathcal{G}$-orbit in $U_0$ is fölner, then all $\mathcal{G}$-orbits are uniformly fölner symmetric.

*Proof.* Consider the notation of Proposition 4.33. Suppose that $\mathcal{G}(x)$ is fölner and dense for some $x \in U_0$. Let $S_n$ be a fölner sequence in $\mathcal{G}(x)$ satisfying the property of the statement of Lemma 5.29. For each $n$, take any $r_n > 0$ such that $\mathrm{Pen}(S_n, 2CK + C + 1) \subset B_E(x, r_n)$. For any $y \in V(x, r_n)$, let $S_{y,n} = \mathrm{Pen}(\phi_{x,y,r_n}(S_n), K)$. Then

$$(5.7) \qquad \#S_n \le \#S_{y,n} \le (\#E)^K \cdot \#S_n$$

because $\phi_{x,y,r_n}$ is injective (Proposition 4.33).

**Claim 13.** We have

$$\partial S_{y,n} \subset \mathrm{Pen}(\phi_{x,y,r_n}(\partial_{2CK+C+1}S_n), K+1).$$

Fix any $z \in \partial S_{y,n}$ to prove this assertion. Then $z \in S_{y,n}$ and $d_E(z, \mathcal{G}(y) \setminus S_{n,y}) = 1$. Thus there is some $z_0 \in \phi_{x,y,r_n}(S_n)$ and some $z_1 \in \mathcal{G}(y) \setminus S_{n,y}$ such that $d_E(z, z_0) \le K$, $d_E(z, z_1) = 1$ and $d_E(z_1, \phi_{x,y,r_n}(S_n)) > K$, yielding $d_E(z, z_0) = K$ and $d_E(z_1, z_0) = K+1$. Let $\bar{z}_0 = \phi_{x,y,r_n}^{-1}(z_0) \in S_n$. Then

$$d_E(y, z_1) \le d_E(y, z_0) + d_E(z_0, z_1) \le d_E(x, \bar{z}_0) + K + 1 \le \frac{r_n}{C}$$

because $\phi_{x,y,r_n}$ is non-expanding (Proposition 4.33). Since $B_E(y, r_n/C) \cap \mathrm{im} \phi_{x,y,r_n}$ is a $K$-net in $B_E(y, r_n/C)$ (Proposition 4.33), it follows that there is some $z_2 \in B_E(y, r_n/C) \cap \mathrm{im} \phi_{x,y,r_n}$ such that $d_E(z_1, z_2) \le K$, and thus

$$d_E(z_2, \phi_{x,y,r_n}(S_n)) \ge d_E(z_1, \phi_{x,y,r_n}(S_n)) - d_E(z_1, z_2) > 0,$$
So $z_2 \notin \phi_{x,y,r_n}(S_n)$ and $\bar{z}_2 = \phi_{x,y,r_n}^{-1}(z_2) \in B_E(x, r_n) \setminus S_n$. Since $\phi_{x,y,r_n}$ is bi-lipschitz with bi-lipschitz constant $C$ (Proposition 4.33), and because

$$d_E(z_0, z_2) \leq d_E(z_0, z_1) + d_E(z_1, z_2) \leq 2K + 1,$$

it follows that

$$d_E(\bar{z}_0, \bar{z}_2) \leq C d_E(z_0, z_1) \leq C(2K + 1).$$

Therefore

$$\bar{z}_2 \in \mathrm{Pen}(S_n, C(2K + 1)) \setminus S_n \subset \partial_{2CK+C+1}S_n.$$

Then

$$z \in \mathrm{Pen}(\phi_{x,y,r_n}(\partial_{2CK+C+1}S_n), K+1)$$

because

$$d_E(z, z_2) \leq d_E(z, z_1) + d_E(z_1, z_2) \leq 1 + K,$$

which finishes the proof of Claim 13.

**Claim 14.** We have

$$\phi_{x,y,r_n}(\mathrm{Pen}(S_n, CK + 1) \setminus \mathrm{Pen}(S_n, CK)) \subset \partial_{CK-K+2}S_{y,n}.$$

First observe that

$$\phi_{x,y,r_n}(\mathrm{Pen}(S_n, CK + 1)) \subset \mathrm{Pen}(\phi_{x,y,r_n}(S_n), CK + 1) = \mathrm{Pen}(S_{y,n}, CK - K + 1)$$

because $\phi_{x,y,r_n}$ is non-expanding (Proposition 4.33). On the other hand, for any $z \in S_{y,n} \cap \mathrm{im} \phi_{x,y,r_n}$, if $\bar{z} = \phi_{x,y,r_n}^{-1}(z)$, then

$$d_E(\bar{z}, S_n) \leq C d_E(z, \phi_{x,y,r_n}(S_n)) \leq CK$$

because $\phi_{x,y,r_n}$ is bi-lipschitz with bi-lipschitz constant $C$ (Proposition 4.33), yielding

$$z \in \phi_{x,y,r_n}(\mathrm{Pen}(S_n, CK)).$$

Hence

$$S_{y,n} \cap \mathrm{im} \phi_{x,y,r_n} \subset \phi_{x,y,r_n}(\mathrm{Pen}(S_n, CK)).$$

Therefore

$$\phi_{x,y,r_n}(\mathrm{Pen}(S_n, CK + 1) \setminus \mathrm{Pen}(S_n, CK)) \subset \mathrm{Pen}(S_{y,n}, CK - K + 1) \setminus S_{y,n} \subset \partial_{CK-K+2}S_{y,n}$$

as desired because $\phi_{x,y,r_n}$ is injective (Proposition 4.33).

On the one hand, from Claim 13, Proposition 4.33 and (3.10), it follows that

$$\begin{align} \#\partial S_{y,n} &\leq \#\mathrm{Pen}(\phi_{x,y,r_n}(\partial_{2CK+C+1}S_n), K+1) \nonumber \\ &\leq (\#E)^K \cdot \#\phi_{x,y,r_n}(\partial_{2CK+C+1}S_n) \nonumber \\ &= (\#E)^K \cdot \#\partial_{2CK+C+1}S_n \nonumber \\ &\leq (\#E)^{2CK+C+K+1} \cdot \#\partial S_n; \end{align}$$
and on the other hand, from Proposition 4.33, Claim 14 and (3.10), it follows that

$$ (5.9) \qquad \begin{aligned} \#(\mathrm{Pen}(S_n, CK + 1) \setminus \mathrm{Pen}(S_n, CK)) &= \#\phi_{x,y,r_n}(\mathrm{Pen}(S_n, CK + 1) \setminus \mathrm{Pen}(S_n, CK)) \\ &\le \#\partial_{CK-K+2}S_{y,n} \\ &\le (\#E)^{CK-K+2} \cdot \#\partial S_{y,n}. \end{aligned} $$

Now, each dense $\mathcal{G}$-orbit $\mathcal{O}$ contains a sequence of points $y_{\mathcal{O},n}$ with $y_{\mathcal{O},n} \in V(x, r_n)$ for all $n$, and then the sets $S_{\mathcal{O},n} = S_{y_{\mathcal{O},n},n}$ form a fölner sequence in $\mathcal{O}$ so that the order of convergence to zero of $\#\partial S_{\mathcal{O},n}/\#S_{\mathcal{O},n}$ is independent of $\mathcal{O}$ by (5.7), (5.8), (5.9) and Lemma 5.29.

When $\mathcal{G}$ is minimal, there is some $K_n > 0$ such that $\mathcal{O} \cap V(x, r_n)$ is a $K_n$-net in every $\mathcal{G}$-orbit $\mathcal{O}$ by Lemma 3.23. Choose a double sequence $y_{\mathcal{O},m,n}$ for each $\mathcal{G}$-orbit $\mathcal{O}$ so that

$$ \mathcal{O} \cap V(x, r_n) = \{y_{\mathcal{O},m,n} \mid m \in \mathbb{Z}_+\}. $$

Hence the sets $S_{\mathcal{O},m,n} = S_{y_{\mathcal{O},m,n},n}$ form a fölner sequence in $\mathcal{O}$ for each $m$ so that the order of convergence to zero of $\#\partial S_{\mathcal{O},n}/\#S_{\mathcal{O},n}$ is independent of $\mathcal{O}$ and $m$ by (5.7), (5.8), (5.9) and Lemma 5.29, and moreover $\bigcup_m S_{\mathcal{O},m,n}$ is a $K_n$-net in $\mathcal{O}$ for every $m$. Therefore all $\mathcal{G}$-orbits are uniformly fölner symmetric. $\square$

Theorem 5.30 has the following direct consequence.

**Theorem 5.31.** Let $(X, \mathcal{F})$ be a compact foliated space. If $\mathcal{F}$ has a dense amenable leaf without holonomy, then all dense leaves are uniformly amenable. If $\mathcal{F}$ is minimal and has a dense fölner leaf without holonomy, then all leaves have uniformly symmetric amenability.

**Remark.** In the $C^1$ case, Theorem 5.31 is elementary by Proposition 2.23 and Lemma 4.27.
# Chapter 6

# Compactifications

## 6.1 Generalities on Compactifications

This section describes the structure of certain compactification of a space which becomes relevant in the study of the asymptotic geometry of the leaves.

**Definition 6.1.** A *compactification* of a topological space *M* is a pair (*Y*, *e*) consisting of a compact topological space *Y* and an embedding *e* : *M* → *Y* of *M* into *Y* as an open and dense subset.

Compactifications of a space $M$ correspond to closed subalgebras of $C^*(M)$ (bounded continuous functions $M \to \mathbb{R}$ with the supremum norm) that contain the constants and generate the topology. The correspondence associates to a compactification $Y$ of $M$ the algebra of bounded continuous functions on $M$ which extend continuously to $Y$. Conversely, if $\mathcal{A}$ is an algebra of continuous bounded functions on $M$ that contains the constants and generates the topology (that is, compact sets can be separated from points by functions in the algebra), then for each $f \in \mathcal{A}$ let $I_f = [\inf f, \sup f]$ and the compactification of $M$ is the closure of the image of $M$ in the compact space $\prod_{f \in \mathcal{A}} I_f$ via the evaluation map $x \in M \mapsto (f(x))_{f \in \mathcal{A}}$.

From the algebraic point of view, a compact Hausdorff space $X$ is determined by the algebraic structure of $C_b(X)$, the algebra of continuous functions on $X$ with the supremum norm topology. The space $X$ is homeomorphic to the maximal ideal space of the algebra $C_b(X)$. Moreover, a homomorphism of algebras $C^*(X) \to C^*(Y)$ induced a continuous map of topological spaces $Y \to X$.

*********************
## 6.2 The Higson Compactification

A significant example of coarse structure is induced by any compactification<fnref n="1" /> $\overline{X}$ of a locally compact space $X$, with boundary $\partial X = \overline{X} \setminus X$. This coarse structure on $X$ is defined by declaring that a subset $E \subset X \times X$ is controlled when

$$ \overline{E} \cap \partial(X \times X) \subset \Delta_{\partial X} $$

in $\overline{X} \times \overline{X}$, where

$$ \partial(X \times X) = (\partial X \times \overline{X}) \cup (\overline{X} \times \partial X). $$

This is called the *topological*<fnref n="2" /> coarse structure associated to the given compactification; it is always proper.

A compactification $\overline{X}$ of a proper coarse space $X$ is said to be a *coarse compactification*, with *coarse corona* $\partial X = \overline{X} \setminus X$, if the identity map from $X$ with its given coarse structure to $X$ equipped with the topological coarse structure arising from $\overline{X}$ is a coarse map. Roughly speaking, this means that the slices of any controlled subset of $X \times X$ “become small when approaching the boundary” $\partial X$; in particular, this holds for the sets of any uniformly bounded family in $X$.

Coarse compactifications of a proper coarse space $X$ can be described in terms of functions as follows. Let $\mathcal{B}(X)$ be the $C^*$ algebra of all bounded $X \to \mathbb{R}$ with the supremum norm, and let $\mathcal{B}_0(X)$ be the $C^*$ subalgebra of all functions $f \in \mathcal{B}(X)$ that vanish at infinity; i.e., such that, for any $\varepsilon > 0$, there is some compact subset $K \subset X$ so that $|f(x)| < \varepsilon$ for all $x \in X \setminus K$. For any $f \in \mathcal{B}(X)$ and every controlled subset $E \subset X \times X$, let the *E-gradient* of $f$ be the function $\nabla_E f \in \mathcal{B}(X)$ defined by

$$ \nabla_E f(x) = \sup\{|f(x) - f(y)| \mid (x, y) \in E\}. $$

A function $f \in \mathcal{B}(X)$ will be called a *Higson function* when $\nabla_E f \in \mathcal{B}_0(X)$ for all controlled subsets $E \subset X \times X$. The set $\mathcal{B}_\nu(X)$ of all Higson functions on $X$ is a $C^*$ subalgebra of $\mathcal{B}(X)$ [?], [?]. When only bounded continuous functions are considered, the notation $C_b(X)$, $C_0(X)$ and $C_\nu(X)$ will be used instead of $\mathcal{B}(X)$, $\mathcal{B}_0(X)$ and $\mathcal{B}_\nu(X)$, respectively. The following lemma shows that Higson functions are preserved by coarse maps.

The terms $\overline{X}$-close maps, $\overline{X}$-controlled sets and $\overline{X}$-coarse compactification will be used in the case of the topological coarse structure induced by a compactification $\overline{X}$ of a locally compact space $X$.

**Lemma 6.2.** Let $\overline{X}$ be a compactification of a locally compact space $X$ with boundary $\partial X$. Then the following conditions are equivalent for any subset $E \subset X \times X$:

(i) $E$ is $\overline{X}$-controlled.

<footnote n="1">¹Only metrizable compactifications are considered in [?], but this kind of coarse structures can be defined for arbitrary compactifications [?].</footnote>

<footnote n="2">²This term is used in [?]. It is also called *continuously controlled* coarse structure in [?].</footnote>
(ii) $\nabla_E f \in \mathcal{B}_0(X)$ for every $f \in \mathcal{B}(X)$ with an extension $\bar{f} : \bar{X} \to \mathbb{R}$ which is continuous on the points of $\partial X$.

(iii) $\nabla_E f \in C_0(X)$ for every $f \in C_b(X)$ with a continuous extension to $\bar{X}$.

*Proof.* To prove that property (i) implies property (ii), suppose that $E$ is $\bar{X}$-controlled, and assume that some $f \in \mathcal{B}(X)$ has an extension $\bar{f} : \bar{X} \to \mathbb{R}$ which is continuous on the points of $\partial X$. Since the function $(x, y) \mapsto |\bar{f}(x) - \bar{f}(y)|$ on $\bar{X} \times \bar{X}$ vanishes on $\Delta_{\partial X}$ and is continuous on the points of $\partial X \times \partial X$, there is some open neighbourhood $\Omega$ of $\Delta_{\partial X}$ in $\bar{X} \times \bar{X}$ such that $|\bar{f}(x) - \bar{f}(y)| < \varepsilon$ for all $(x, y) \in \Omega$. On the other hand, since $E$ is $\bar{X}$-controlled, there is some open neighbourhood $U$ of $\partial X$ in $\bar{X}$ such that

$$ \bar{E} \cap (U \times \bar{X}) \subset \Omega. $$

It follows that, for the compact subset $K = X \setminus U$, we have $\nabla_E f(x) < \varepsilon$ for all $x \in X \setminus K = X \cap U$. Hence $\nabla_E f \in \mathcal{B}_0(X)$.

Property (iii) is a particular case of property (ii).

To prove that property (iii) implies property (i), assume that $\nabla_E f \in C_0(X)$ for all $f \in C_b(X)$ with a continuous extension to $\bar{X}$. If $E$ were not $\bar{X}$-controlled, there would be a pair of different points, $x \in \partial X$ and $y \in \bar{X}$, such that either $(x, y)$ or $(y, x)$ is in $\bar{E}$. Since the family of controlled sets is invariant by transposition [?], [?], we can indeed suppose that $(x, y) \in \bar{E}$. Then, for any continuous function $\bar{f} : \bar{X} \to \mathbb{R}$ with $\bar{f}(x) \neq \bar{f}(y)$, the restriction $f = \bar{f}|_X$ would satisfy

$$ \liminf_{z \to x} \nabla_E f(z) \geq |\bar{f}(x) - \bar{f}(y)| > 0, $$

which would be a contradiction. Therefore $E$ is $\bar{X}$-controlled. $\square$

<figure><img src="image_10.png" /></figure>

The following is a direct consequence of Lemma 6.2.

**Corollary 6.3.** Let $\bar{X}$ be a compactification of a proper coarse space $X$ with boundary $\partial X$. Then the following conditions are equivalent:

(i) $\bar{X}$ is a coarse compactification of $X$.

(ii) $\mathcal{B}_\nu(X)$ contains every function in $\mathcal{B}(X)$ with an extension to $\bar{X}$ which is continuous on the points of $\partial X$.

(iii) $C_\nu(X)$ contains every continuous function $X \to \mathbb{R}$ that extends continuously to $\bar{X}$.

**Proposition 6.4.** Let $\bar{X}$ and $\bar{X}'$ be compactifications of locally compact spaces $X$ and $X'$ with boundaries $\partial X$ and $\partial X'$, respectively. Then the following properties hold:

(i) A map $\phi : X \to X'$ is coarse if it has an extension $\bar{\phi} : \bar{X} \to \bar{X}'$ which is continuous on the points of $\partial X$ and such that $\bar{\phi}(\partial X) \subset \partial X'$.

(ii) Let $\phi, \psi : X \to X'$ maps with extensions $\bar{\phi}, \bar{\psi} : \bar{X} \to \bar{X}'$ satisfying the conditions of property (i). Then $\phi$ and $\psi$ are $\bar{X}'$-close if and only if $\bar{\phi} = \bar{\psi}$ on $\partial X$.
Proof. Let $\phi : X \to X'$ be a map satisfying the conditions of property (i). Let $B$ be a bounded subset of $X'$, which means that it has compact closure in $X'$, and thus $\bar{B} \cap \partial X' = \emptyset$. So

$$ \bar{\phi}(\bar{\phi}^{-1}(\bar{B}) \cap \partial X) \subset \bar{\phi}(\bar{\phi}^{-1}(\bar{B})) \cap \partial X' \subset \bar{B} \cap \partial X' = \emptyset $$

because $\bar{\phi}$ is continuous on the points of $\partial X$. It follows that $\bar{\phi}^{-1}(\bar{B}) \cap \partial X = \emptyset$, and thus $\phi^{-1}(B)$ has compact closure in $X$; that is, $\phi^{-1}(B)$ is bounded in $X$.

Let $E$ be a controlled subset of $X \times X$, and let $f : X' \to \mathbb{R}$ be a bounded function with an extension $\bar{f}$ to $\bar{X}'$ which is continuous on the points of $\partial X'$. Since $\bar{f} \circ \bar{\phi}$ is continuous on the points of $\partial X$ and extends the function $f \circ \phi$, we get $\nabla_{(\phi \times \phi)(E)} f = \nabla_E (f \circ \phi) \in \mathcal{B}_0(X)$ by Lemma 6.2. It follows that $(\phi \times \phi)(E)$ is a controlled subset of $X' \times X'$ by Lemma 6.2. Therefore $\phi$ is a coarse map, which shows property (i).

Let $\phi, \psi : X \to X'$ be maps with extensions $\bar{\phi}, \bar{\psi} : \bar{X} \to \bar{X}'$ satisfying the conditions of property (i). Suppose first that $\bar{\phi} = \bar{\psi}$ on $\partial X$, and let $E = \{(\phi(x), \psi(x)) \mid x \in X\}$. Fix any point $(x', y') \in \bar{E} \cap \Delta_{\partial(X' \times X')}$. Thus, for each neighbourhood $\Omega$ of $(x', y')$ in $\bar{E}$, there is some point $x_{\Omega} \in X$ so that $(\phi(x_{\Omega}), \psi(x_{\Omega})) \in \Omega$; such points $x_{\Omega}$ form a net $(x_{\Omega})$ in $X$. Suppose e.g. that $x' \in \partial X'$. Then the net $(\phi(x_{\Omega}))$ is unbounded in $X'$, and thus the net $(x_{\Omega})$ is unbounded in $X$ because $\phi$ is a coarse map according to property (i). So there is an accumulation point $x$ of $(x_{\Omega})$ in $\partial X$. Since $\bar{\phi}$ and $\bar{\psi}$ are continuous at $x$, it follows that $(\bar{\phi}(x), \bar{\psi}(x))$ is an accumulation point of the net $(\phi(x_{\Omega}), \psi(x_{\Omega}))$, which converges to $(x', y')$. Hence $(x', y') = (\bar{\phi}(x), \bar{\psi}(x)) \in \Delta_{\partial X'}$ because $\bar{\phi} = \bar{\psi}$ on $\partial X$. This shows that $E$ is $\bar{X}'$-controlled, and thus $\phi$ is $\bar{X}'$-close to $\psi$.

Now assume that $\phi$ is $\bar{X}'$-close to $\psi$; i.e., the set $E = \{(\phi(x), \psi(x)) \mid x \in X\}$ is $\bar{X}'$-controlled. By the conditions satisfied by $\bar{\phi}$ and $\bar{\psi}$, we get

$$ \begin{aligned} (\bar{\phi} \times \bar{\psi})(\Delta_{\partial X}) &= (\bar{\phi} \times \bar{\psi})(\overline{\Delta_X} \cap (\partial X \times \partial X)) \\ &\subset (\overline{\phi \times \psi})(\Delta_X) \cap (\partial X' \times \partial X') \\ &= \overline{E} \cap (\partial X' \times \partial X') \\ &\subset \Delta_{\partial X'} \end{aligned} $$

yielding $\bar{\phi} = \bar{\psi}$ on $\partial X$, which completes the proof of property (ii). $\square$

<figure><img src="image_11.png" /></figure>

According to Corollary 6.3, there is a maximal coarse compactification $X^{\nu}$, which is the maximal ideal space of $C_{\nu}(X)$; it is called the *Higson compactification* of $X$, and its boundary $\nu X$ is called the *Higson corona*. Note that $X^{\nu}$ is the maximal ideal space of $\mathcal{B}_{\nu}(X)$ as well because each Higson function on $X$ has a unique extension to $X^{\nu}$ which is continuous on the points of $\nu X$. So there are canonical isomorphisms

$$ (6.1) \qquad C(\nu X) \cong C_{\nu}(X)/C_0(X) \cong \mathcal{B}_{\nu}(X)/\mathcal{B}_0(X). $$

For subsets $A$ of $X$ or of $X \times X$, the notation $A^{\nu}$ will be used to indicate the closure of $A$ in $X^{\nu}$ or in $X^{\nu} \times X^{\nu}$, respectively. The notation $\nu(X \times X) = (\nu X \times X) \cup (X \times \nu X)$ will be also used.
**Lemma 6.5.** Let $\phi: X \to X'$ be a coarse map between proper coarse spaces. Then:

(i) $f \circ \phi \in \mathcal{B}_0(X)$ for all $f \in \mathcal{B}_0(X')$, and

(ii) $f \circ \phi \in \mathcal{B}_\nu(X)$ for all $f \in \mathcal{B}_\nu(X')$.

*Proof.* Property (i) is a direct consequence of the hypotheses.

Let $f \in \mathcal{B}_\nu(X')$. If $E$ is a controlled subset of $X \times X$, then $E' = (\phi \times \phi)(E)$ is a controlled subset of $X' \times X'$ because $\phi$ is coarse. Hence $\nabla_{E'}f \in \mathcal{B}_0(X')$, yielding that $(\nabla_{E'}f) \circ \phi \in \mathcal{B}_0(X)$. Then property (ii) follows because $(\nabla_{E'}f) \circ \phi = \nabla_E(f \circ \phi)$. $\square$

**Proposition 6.6.** Any coarse map $\phi: X \to X'$ between proper coarse spaces has a unique extension $\bar{\phi}: X^\nu \to X'^\nu$ which is continuous on the points of $\nu X$ and such that $\bar{\phi}(\nu X) \subset \nu X'$.

*Proof.* According to Lemma 6.5, $\phi$ induces a homomorphism $\phi^* : \mathcal{B}_\nu(X') \to \mathcal{B}_\nu(X)$ defined by $\phi^*(f) = f \circ \phi$, which maps $\mathcal{B}_0(X')$ to $\mathcal{B}_0(X)$. By (6.1), $\phi^*$ induces a homomorphism $C(\nu X') \to C(\nu X)$. Then, by considering maximal ideal spaces, we get a map $\bar{\phi}: X^\nu \to X'^\nu$, which extends $\phi$ and maps $\nu X$ to $\nu X'$. The continuity of $\bar{\phi}$ on the points of $\nu X$ is a consequence of the fact that any Higson function has a unique extension to the Higson compactification which is continuous on the Higson corona. $\square$

Some times the Higson compactification can be easily determined, as shown by the following result.

**Proposition 6.7.** Let $X$ be a proper coarse space with the topological coarse structure induced by a first-countable compactification $\bar{X}$ of $X$ with boundary $\partial X$. Then $\bar{X}$ and $X^\nu$ are equivalent compactifications of $X$, and thus $\partial X$ is homeomorphic to $\nu X$.

*Proof.* Since $\bar{X}$ is a $\bar{X}$-coarse compactification of $X$, it turns out that the identity map on $X$ extends to a continuous map $\pi: X^\nu \to \bar{X}$. Suppose that $\pi$ is not injective; thus we have $\pi(x) = \pi(y)$ for some different points $x, y \in \nu X$. Let $P$ and $Q$ be disjoint compact neighbourhoods of $x$ and $y$ in $X^\nu$. Then the closures of $P \cap X$ and $Q \cap X$ in $\bar{X}$ contain $\pi(x)$. By the first-countablity of $\bar{X}$, there are sequences, $x_n$ in $P \cap X$ and $y_n$ in $Q \cap X$, which converge to $\pi(x)$ in $\bar{X}$. Let $E = \{(x_n, y_n) | n \in \mathbb{Z}_+\}$, and let $\bar{E}$ and $E^\nu$ denote the closures of $E$ in $\bar{X}$ and $X^\nu$, respectively. We have

$$ \bar{E} \cap \partial(X \times X) = \{(\pi(x), \pi(y))\} \subset \Delta_{\partial X}, $$

whilest $E^\nu \subset P \times Q$, which does not meet $\Delta_{\nu X}$. Thus $E$ is $\bar{X}$-controlled, but it is not $X^\nu$-controlled, which contradicts the fact that $X^\nu$ is a $\bar{X}$-coarse compactification. $\square$

Even if the statement of Proposition 6.7 were not true when the first-countability axiom is removed, we have the following result, which follows easily from Lemma 6.2 and Corollary 6.3.

**Proposition 6.8.** Topological coarse structures are induced by their Higson compactifications.
The following is a direct consequence of Propositions 6.4, 6.6 and 6.8.

**Corollary 6.9.** Let $X$ and $X'$ be locally compact spaces with topological coarse structures. Then the following properties hold:

(i) A map $\phi: X \to X'$ is coarse if and only if it has an extension $\phi^\nu: X^\nu \to X'^\nu$ which is continuous on the points of $\nu X$ and such that $\phi^\nu(\nu X) \subset \nu X'$.

(ii) Two coarse maps $\phi, \psi: X \to X'$ are close if and only if the extensions $\phi^\nu$ and $\psi^\nu$, given by property (i), are equal on $\nu X$.

**Corollary 6.10.** Let $X$ and $X'$ locally compact spaces with topological coarse structures. Then a map $\phi: X \to X'$ is a coarse equivalence if and only if it has an extension $\phi^\nu: X^\nu \to X'^\nu$ such that $\phi^\nu(\nu X) \subset \nu X'$, $\phi^\nu$ is continuous on the points of $\nu X$ and the restriction $\phi^\nu: \nu X \to \nu X'$ is a homeomorphism.

The following result shows that proper metric coarse structures are particular cases of the topological ones.

**Proposition 6.11.** The metric coarse structure of each proper metric space is equal to the topological coarse structure induced by its Higson compactification.

*Proof.* Let $M$ be a proper metric space with metric $d$. Since $M^\nu$ is a coarse compactification of $M$, it is enough to prove that any $M^\nu$-controlled subset $E \subset M \times M$ is also metric controlled. Suppose that $E$ is not metric controlled. This means that there is a sequence of points $(x_n, y_n)$ in $E$ such that $d(x_n, y_n) \to \infty$. In particular, one of these sequences, say e.g. $x_n$, goes to infinity; that is, for any compact subset $K \subset M$, there is some positive integer $N$ so that $x_n \notin K$ for all $n \ge N$. Take any sequence $r_n \uparrow \infty$ so that $0 < r_n \le d(x_n, y_n)$ for all $n$, and the balls $B_M(x_n, r_n)$ are disjoint from each other. Then a continuous function $f: M \to \mathbb{R}$ is well defined by

$$f(x) = \begin{cases} \frac{r_n - d(x_n, x)}{r_n} & \text{if } d(x_n, x) \le r_n \\ 0 & \text{otherwise.} \end{cases}$$

Observe that $f \in C_\nu(M)$ while $\nabla_E f(x_n) \ge |f(x_n) - f(y_n)| = 1$ for all $n$, yielding that $\nabla_E f \notin C_0(M)$, and thus $E$ is not $M^\nu$-controlled by Lemma 6.2, which is a contradiction. $\square$

Proposition 6.11 and Corollary 6.10 have the following consequences.

**Corollary 6.12.** Let $M$ and $M'$ locally compact metric spaces. Then a map $\phi: M \to M'$ is a coarse equivalence if and only if it has an extension $\phi^\nu: M^\nu \to M'^\nu$ such that $\phi^\nu(\nu M) \subset \nu M'$, $\phi^\nu$ is continuous on the points of $\nu M$ and the restriction $\phi^\nu: \nu M \to \nu M'$ is a homeomorphism.

According to Corollary 3.16, ***
## 6.3 The Higson Compactification of a Metric Space

The compactification to be studied in this section, called the Higson compactification, will be described by its associated algebra of functions. The following definition will be required.

**Definition 6.13.** The variation of a function $f: M \to \mathbb{C}$ on a metric space $M$ is the function

$$V_r f(x) = \sup\{|f(y) - f(x)| \mid y \text{ in } B(x, r)\}.$$

The variation of a function has several elementary properties that will be used below. The proof of these properties is completely elementary.

**Lemma 6.14.** Let $f, g$ be two bounded functions on the metric space $M$. Then

$$1. \quad V_r f \le 2 \sup |f|$$

$$2. \quad V_r(f+g) \le V_r f + V_r g$$

$$3. \quad V_r(fg) \le \sup |g| V_r f + \sup |f| V_r g$$

$$4. \quad V_r f \le V_s f \text{ if } r < s.$$

**Proposition 6.15.** Let $M$ be a metric space and let $C_\nu(M)$ be the collection of bounded continuous functions $f$ on $M$ such that $V_r f$ vanishes at $\infty$ on $M$, for all $r > 0$. Then $C_\nu(M)$ is a closed ideal of $C_b(M)$ in the uniform topology.

*Proof.* The property that $C_\nu(M)$ is a subalgebra of the algebra of bounded, continuous functions on $M$ follows immediately from Lemma 6.14 (c). It is also an ideal because of the related inequality $V_r(fg) \le \|g\|V_r(f)$ that is valid for any $g$ in $C_b(M)$ and any $f \in C_\nu(M)$.

To show that $C_\nu(M)$ is closed in the uniform topology, let $\{f_n\}$ be a sequence in $C_\nu(M)$ that converges uniformly to $f$, a bounded continuous function on $M$. Then

$$V_r(f)(x) \le 2\|f - f_n\| + V_r(f_n)(x)$$

for all $n$ and all $x$ in $M$. Given $\varepsilon > 0$, choose $n$ so large that $\|f - f_n\| < \varepsilon/3$, and then choose a compact subset $K$ of $M$ such that $V_r(f_n) < \varepsilon/3$ on $M \setminus K$. It follows that $V_r(f) < \varepsilon$ on $M \setminus K$, and thus that $f \in C_\nu(M)$. $\square$

The algebra $C_\nu(M)$ always contains the constant functions. If furthermore $M$ is locally compact, then the compactly supported functions belong to $C_\nu(M)$, and thus this algebra generates the topology of $M$. Hence, in the case the $M$ is locally compact, the algebra $C_\nu(M)$ determines a compactification of $M$.

**Definition 6.16.** The Higson compactification of a locally compact metric space $M$ is the compact hausdorff space whose algebra of continuous functions is $C_\nu(M)$.

**Remark.** The Higson compactification makes sense for locally compact uniform spaces.
**Remark.** If $f$ is a function on a metric space $(M, d)$, define the length of the gradient of $f$ by

$$|\nabla f|(x) = \limsup_{y \to x} \frac{|f(y) - f(x)|}{d(y, x)}.$$

If $M$ is a path metric space, and $f$ is a continuous function $f : M \to \mathbb{C}$ such that $|\nabla f|(x) \to \infty$ as $x$ diverges to infinity in $M$, then $f$ is in $C_\nu(M)$. This is so because if $x$ and $y$ are in $M$, and $y$ is at distance $\le r$ from $x$, then

$$|f(x) - f(y)| \le \int_{0}^{d(x,y)} |\nabla f|(\gamma(t)) \cdot dt,$$

where $\gamma : [0, d(x, y)] \to M$ is a geodesic from $x$ to $y$. Thus if $|\nabla f| \to 0$ at infinity, then also $V_r f \to 0$ at infinity for all $r > 0$.

The collection of functions $f$ on $M$ with the property that the gradient $|\nabla f|$ vanishes at infinity is an algebra which contains the constant functions and generated the topology. The closure of this algebra in $C(M)$ is denoted by $C_\nabla(M)$.

There is a norm preserving inclusion $C_\nabla(M) \hookrightarrow C_\nu(M)$ showing that the Higson compactification dominates the compactification associated to $C_\nabla(M)$.

One compactification that has been frequently used in the theory of foliations is the endpoint (or Freudental) compactification. For a metric space $(M, d)$, this corresponds to the algebra of bounded continuous functions $f$ on $M$ such that $|\nabla f| \equiv 0$ outside a compact subset of $M$. This algebra contains the constant functions, and also the compactly supported functions, so it generated the topology of $M$ when $M$ is locally compact.

The collection of functions $f$ on $M$ such that $V_r f = 0$ outside a compact subset of $M$ is precisely the collection of continuous functions which are constant outside a compact subset. This family of functions forms a subalgebra of $C(M)$ and its closure gives rise to the endpoint or Freudenthal compactification of $M$.

The endpoint compactification of the leaves of a foliated space has been proven relevant for questions related to dynamics and to generic topology. It is however somewhat smaller for understanding the finer questions related to generic geometry and quasi-isometry invariant of leaves.

**Definition 6.17.** A function $f$ from $M$ into a uniform space $Y$ is *slowly oscillating* if given $r > 0$ and an entourage $U$ of the uniformity of $Y$ there is a compact set $K_{U,r}$ in $M$ such that $(f(x), f(y)) \in U$ whenever $d(x, y) < r$ and $x, y \in M \setminus K_{U,r}$. In particular, when $Y = \mathbb{R}$, this says that for each $r > 0$, the function $x \in M \mapsto \text{diam} f(B(x, r))$ tends to 0 as $x \to \infty$ in $M$.

There is a characterization of the Higson compactification (due to Keesling [18]) which resembles the familiar characterization of the Stone-Čech compactification. Namely, if $M$ is a proper metric space, then the Higson compactification of $M$ is the unique compactification having the following property: if $Y$ is a compact (completely regular) uniform space and $f : M \to Y$ is slowly oscillating, then $f$ extends to $M^\nu$.
Another compactification that is relevant for questions of quasi-isometries is the Royden compactification. This requires that the metric space $M$ be also endowed with a borel measure $\mu$. Then the collection of bounded continuous functions $f: M \to \mathbb{C}$ such that $\int_M |\nabla f|^2 \cdot \mu < \infty$ forms a subalgebra of the algebra of bounded continuous functions, and it also contains the constant functions and the compactly supported functions. It gives rise to the Royden compactification of $M$.

**Theorem 6.18.** Let $M, M'$ be riemannian manifolds, endowed with the induced path metric structure and riemannian measure. Then, if $\dim M = \dim M' > 2$, $M$ and $M'$ are quasi-isometric if and only if the Royden compactifications are homeomorphic. In dimension 2, homeomorphism of the Royden compactification is equivalent to the surfaces being quasi-conformally equivalent.

This result has contributions of several authors, including Nakai [?], Ferrand [?]. There is no such result for the Higson compactifications of general metric spaces, as the following example will show. Nevertheless, in the category of coarsely convex metric spaces it is still true that an algebraic isomorphism of Higson algebras implies a quasi-isometric homeomorphism of the underlying metric spaces.

**Example 6.19.** Let $\mathbb{N}$ be the integers with the standard metric, and let $\mathbb{N}^2 = \{n^2 \mid n \in \mathbb{N}\}$ and $\mathbb{N}^3 = \{n^3 \mid n \in \mathbb{N}\}$, both with the metric induced from $\mathbb{N}$. Every function $f$ on $\mathbb{N}^2$ and on $\mathbb{N}^3$ satisfies $V_r f(x) \to 0$ as $x \to \infty$; therefore the Higson compactifications of $\mathbb{N}^2$ and $\mathbb{N}^3$ are homeomorphic to their respective Stone-Cech compactification, and the natural map $n^2 \mapsto n^3$ of $\mathbb{N}^2$ onto $\mathbb{N}^3$ induces a homeomorphism of compactifications. The spaces $\mathbb{N}^2$ and $\mathbb{N}^3$ 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 $S = R^3$.

Before describing some aspects of the Higson compactification of the leaves of a foliated space, one more definition is required.

**Definition 6.20.** If $(Y, e)$ is a compactification of a space $M$, then the compact space $Y \setminus e(M)$ is called the corona of $M$ relative to $(Y, e)$. In particular, $\nu M = M^\nu \setminus M$ is called the Higson corona of $M$.

****

The following results examine the topological structure of the Higson compactification.

**Proposition 6.21.** Let $(M, d)$ a proper metric space. If $U$ is a neighborhood of a point $p$ in $\nu M$, then $U \cap M$ contains metric balls of arbitrarily large radius and whose centers have $p$ as an accumulation point in $\nu M$. Conversely, if $W$ is a subset of $M$ which contains metric balls of arbitrarily large radius, then the closure of $W$ in $M^\nu$ is a neighborhood of a point in $\nu M$, and this point is an accumulation point of the centers of those metric balls.

*Proof.* If $p$ is a point in $M^\nu$ and $U$ is a neighborhood of $p$, then there is a continuous function $F: M^\nu \to [0, 1]$ such that $F(p) = 0$ and $F(\nu M \setminus U) = 1$. If $f$ is the restriction of $F$ to $M$,
then $V_r f \to 0$ as $x \to \infty$ in $M$, for each $r > 0$. Fix $r > 0$; if $0 < \varepsilon < 1$, there is a compact subset of $M$ on whose complement the variation $V_r f < \varepsilon/2$. For a point $x$ in $M \cap U$ outside of that compact set, the inequality $V_r f(x) < \varepsilon/2$ implies that $|f(y) - f(x)| < \varepsilon/2$. Because $f$ extends continuously to $\nu M$ and equals 0 at $p$, the point $x$ may be also chosen so that $|f(x)| < \varepsilon/2$. It follows that $|f| < \varepsilon$ on the ball $B(x, r)$, that is, $B(x, r) \subset U \cap M$. Since $r > 0$ was arbitrary, this proves the first claim.

For the second statement, let $W$ be a subset of $M$ with the property that it contains metric balls of arbitrarily large radius. It has to be shown that the closure $\overline{W}$ of $W$ in $M^\nu$ is a neighborhood of a point in the Higson corona of $M$. Equivalently, that there is a function $F \ge 0$ on $\eta M$ whose support is contained in $U$ and a point $p$ in the Higson corona such that $F(p) > 0$.

Let $x_n$ be a sequence of points in $W$ (without limit point in $X$) and a sequence of radii $r_n \to \infty$ such that $B(x_n, r_n) \subset W$. Assume that the balls $B(x_n, r_n)$ are mutually disjoint, and define $f(x)$ by

$$f(x) = \begin{cases} 1 - \frac{d(x, x_n)}{r_n} & x \text{ in } B(x_n, r_n) \\ 0 & \text{otherwise.} \end{cases}$$

It is clear that $f$ is a bounded continuous function on $M$. If $r > 0$ and $\varepsilon$ are given, chose $N$ so that $r/r_n < \varepsilon$ for all $n \ge N$. Then $V_r f|(x) < \varepsilon$ if $x$ is outside the ball $B(x_j, r_j + r)$, $j = 1, \dots, N$. Therefore $f$ satisfies $V_r f(x) \to 0$ as $x \to \infty$, and so it extends to a continuous function $F$ on $M^\nu$. By construction, the support of $F$ is contained in the closure of $W$ in $M^\nu$. Furthermore, if $p$ is any accumulation point of $x_n$, then $p$ is in the Higson corona and $F(p) = 1$. If follows that $F^{-1}(0, 1]$ is an open subset of $M^\nu$ contained in $W$. $\square$

**Corollary 6.22.** Let $M$ be a locally compact metric space, an let $M^\nu$ be its Higson compactification. A point $p$ in $M^\nu$ is in $M$ if and only if the set $\{p\}$ is a $G_\delta$-set.

*Proof.* The only if part is elementary. The if part is essentially a corollary of the proof of Proposition 6.21. If $p \in M^\nu$ is such that $\{p\}$ is a $G_\delta$ set, then there is a sequence $(x_n)$ in $M$ that converges to $p$. Passing to a subsequence if necessary, it may be assumed that the metric balls $B(x_n, n)$ are mutually disjoint. Let $f: M \to \mathbb{R}$ be the function

$$f(x) = \begin{cases} (-1)^n \frac{r_n - d(x, x_n)}{r_n} & x \text{ in } B(x_n, r_n) \\ 0 & \text{otherwise.} \end{cases}$$

Then $f$ extends to a continuous function $F$ on $M^\nu$, thus $\lim_n F(x_n) = F(p)$. But the definition of $f$ implies $F(x_n) = (-1)^n$, so the limit $\lim_n F(x_n)$ does not exist. $\square$

**Proposition 6.23.** Let $\phi: M \to N$ be a quasi-isometry. Then the induced map $f \mapsto \phi \circ f$ from functions on $N$ to functions on $M$ takes $C_\nu(N)$ into $C_\nu(N)$.

*Proof.* By the definition of quasi-isometry, there are constants $A$ and $B$ such that

$$\frac{1}{A} d_L(x, y) - B \le d_N(\phi x, \phi y) \le Ad_L(x, y) + B$$
for all $x$ and $y$ in $L$. The second inequality implies that the image by $\phi$ of a ball with center $x$ and radius $r$ is contained in the ball of center $\phi x$ and radius $Ar + B$. Therefore:

$$V_r f \circ \phi(x) \leq V_{Ar+B} f(\phi(x)).$$

On the other hand, the first inequality implies that if $x \to \infty$ in $M$, then its image $\phi(x) \to \infty$ in $N$. From these two observations it follows that $V_r f \circ \phi \to 0$ at infinity if the same holds true for $f$. $\square$

**Corollary 6.24.** Under the hypothesis of the previous proposition, $\phi$ induces a continuous map $\phi : M^\nu \to N^\nu$ which embeds*** $\nu M$ into $\nu N$.

Let $N$ be a net in the complete proper metric space $M$. The above corollary says that the inclusion $N \hookrightarrow M$ induces an embedding $\nu N \to \nu M$. Moreover

**Proposition 6.25.** The inclusion $N \hookrightarrow M$ of a net in $M$ induces a homeomorphism between the corresponding Higson coronas.

*Proof.* The proof is from [?]. This will be shown by constructing a homomorphism of the corresponding rings of continuous functions. The ring of continuous functions $C^*(\nu M)$ is isomorphic to the quotient ring $C_\nu(M)/C_0(M)$, where $C_0(M)$ denotes the ring of continuous functions which vanish at infinity on $M$.

Given the net $N$ in $M$, there is a positive number $r > 0$ such that the balls $B(a, r)$, $a \in N$, form a locally finite cover of $M$. Let $\phi_a$ be a partition of unity subordinate to this cover. If $f$ is a function on $N$, let $\Phi f$ denote the function on $L$ defined by

$$\Phi f(x) = \sum_{a \in N} f(a) \phi_a(x).$$

It is plain that the operator $\Phi$ takes functions on $N$ to continuous functions on $L$, and that it preserves the algebraic operations of $C^*(N)$. Moreover,

$$\|\Phi f\| \leq \|f\|.$$

***

$\square$

**Proposition 6.26.** Let $F : M \to N$ be a continuous coarse map. Then $F$ induces a homomorphism of algebras $F^* : C_\nu(N) \to C_\nu(M)$ and therefore a continuous map $F^\nu : M^\nu \to N^\nu$.

*Proof.* For each $R > 0$ there is $S(R)$ such that if $d(x, y) < R$, then $d(F(x), F(y)) < S$. There is no loss in generality if it is assumed that $S(R) \to \infty$ as $R \to \infty$.

For any continuous function $f$ on $N$, $V_r(f \circ F)(x) = \sup\{d(f(F(x)), f(F(y))) | d(x, y) < r\}$. But if $d(x, y) < r$, then $d(F(x), F(y)) < S(r)$. Therefore, $V_r(f \circ F)(x) \leq V_{S(r)}f(F(x))$. If $f$ is also in $C_\nu(N)$, then for each $\varepsilon > 0$ there is a compact set $K$ in $N$ such that $V_{S(r)}f < \varepsilon$ on the complement of $K$. Because $F$ is continuous and coarse, it is also proper; therefore $V_r(f \circ F) < \varepsilon$ on the complement of the compact set $F^{-1}K$. $\square$
**Theorem 6.27.** Let $M$ and $N$ be locally compact metric spaces, and suppose that $\varphi: C_\nu(N) \to C_\nu(M)$ is an algebraic isomorphism of their associated Higson algebras. Then $M$ and $N$ are coarsely equivalent. Furthermore, is $M$ and $N$ are quasi-conves, then $\varphi$ induced a continuous coarse isometry between $M$ and $N$.

*Proof.* An algebraic isomorphism $\varphi: C_\nu(N) \to C_\nu(M)$ is automatically continuous and induces a homeomorphism of Higson compactifications $F: M^\nu \to N^\nu$.

This homeomorphism $F$ must send $M$ homeomorphically onto $N$ because no point in the Higson corona is a $G_\delta$-set (Proposition ??). Suppose that the induced map $F: M \to N$ is not coarse. Then there are sequences $x_n, y_n$ in $M$ such that $d(x_n, y_n) \le R$, but $f(F(x_n), F(y_n)) \ge n$. By passing to subsequences, it may be assumed that $d(x_n, x_m) \ge n$ for all $m < n$ and for all $n$, and similarly for $y_n$, and also for the sequences $F(x_n)$ and $F(y_n)$. Let $f$ be a function in $C_\nu(N)$ constructed as in Proposition ?? and having $f(F(x_n)) = 1$, $f(F(y_n)) = -1$. The function $f \circ F$ does not belong to $C_\nu(M)$ because $V_R(f \circ F)(x_n) \ge 2$ for all $n$. This contradicts that $F^*$ is an isomorphism between the corresponding Higson algebras.

The same arguments applies to the inverse homeomorphism $F^{-1}: N^\nu \to M^\nu$.

Finally, it was previously shown in Proposition ?? that a coarse continuous map between quasi-conves spaces is a coarse quasi-isometry. $\square$

**Example 6.28.** This result implies that a coarse equivalence between two locally compact metric spaces induces a homeomorphism between their respective coronas. The converse is not true in general. In example ?? it was shown that the Higson compactifications of the subspaces $N^2$ and $N^3$ of the natural numbers are the same as their Stone-Cech compactifications (which are homeomorphic to the stone-Cech compactification of the natural numbers).

If the continuum hypothesis is accepted, then the Stone-Cech corona of the natural numbers has $2^c$ automorphism. On the other hand, there are at most $c$ maps of $N$ into $N$. Therefore, many homeomorphism of the Higson corona of $N^2$ into that of $N^3$ are not induced by a map of $N^2$ into $N^3$.

Let $M$ be a locally compact space. We want to find an algebra that characterizes the coarse quasi-isometry type of $M$. Let $N$ be a discrete net in $N$ and $Z$ be its closure in the Higson compactification of $M$. Then by the above, $Z = N \cup \nu M$. Fis a point $x_0$ in $M$ and let $Z_n = \{x \in M \mid d(x, x_0) \ge n\}$. Then $Z_1 \supset Z_2 \supset \dots$ is a decreasing sequence of compact subsets of $M^\nu$ whose intersection is $\nu M$. Since each $N_n$ is discrete, the algebra of continuous functions on $N$ is the same as the algebra of bounded borel functions. Therefore, there is a directed systems of Banach algebras

$$C_\nu(Z_1) \to C_\nu(Z_2) \to \dots$$

The direct limit $C_{\nu,\infty}(Z) = \lim \to C_\nu(Z_n)$ is a Banach algebra and there is a canonical map $C_{\nu,\infty}(Z) \to C(Z)$.

Let $M$ be a topological space and let $U \subset M$ be an open subset. Then $U^-$, the closure of $U$ in $M^\nu$, is a compactification of $U$. The following result examines the relation between the Higson compactification of $U$ and the compactification $U^-$.
**Proposition 6.29.** The Higson compactification of $U$ dominates the compactification $U^-$; that is, there is a continuous map $j: U^\nu \to U^-$ that restricts to the identity on $U$.

*Proof.* If $f$ is a function in $C_\nu(U)$, then $V_f f(x) \to 0$ as $x \to \infty$ on $U$ implies that $f$ extends to a continuous function on $M$ that is constant in the complement of $U$. This function $\tilde{f}$ is in $C_\nu(M)$. Indeed, if $x$ and $y$ are limit points of $U$, and if $x_n \to x$ and $y_n \to y$, then $V_r f(x) \to 0$ as $x \to \infty$ in $U$, for each $r$ implies that. ****

□

## 6.4 Compacifications and Limit Sets

Let $X$ be a foliated space, let $F$ be a leaf of $X$, and let $F^\gamma$ be a compactification of $F$ with corona $\gamma F$.

**Definition 6.30.** The limit set of a point $e$ in the corona $\gamma F$, denoted by $\lim(e)$, is the cluster set of the inclusion mapping $F \hookrightarrow X$ at $e$, that is

$$ \lim(e) = \bigcap_{\{U\}} \mathrm{Cl}_X(U \cap F), $$

where $\{U\}$ denotes the collection of open neighborhoods of $e$ in $F^\gamma$.

The limit set $\lim(e)$ is a closed subset of $X$ which may or may not be saturated. A couple of examples will serve as illustration of this fact.

**Example 6.31.** (1) The simplest compactification of $L$ is the Alexandroff or one-point compactification, which has as limit set the limit set of $L$, which of course is a saturated subset of $X$. The same property is shared by the endpoint compactification.

(2) There are many examples of compactifications with non-saturated limit sets. For instance, take a linear foliation on the three dimensional torus with planar leaves and consider a compactification of the planar leaves by adding a circle at infinity, one point for each direction emanating from the origin $**$. Also, the Stone-Čech compactification of a leaf $L$ has the property of saturated limit sets if and only if it is trivial, i.e., if $L$ is already compact.

(3) Another example is the following. Let $X$ be a compact foliated space with a dense leaf $F$. Then it is possible to construct a foliated space $Y$ which is a compactification of $F$ and whose corona is homeomorphic to $X$. Let $F'$ be a copy of $F$ and let $Y$ be the disjoint union $X \cup L'$ with the following topology. A neighborhood of a point of $L'$ is a manifold neighborhood. A neighborhood of a point in $X$ is of the form $U \cup U'$, where $U$ is a neighborhood of the point in $X$, and $U'$ is the intersection of $U$ and $L$ in $X$, but viewed in the copy $L'$. In terms of algebras of functions, it is described by the continuous functions on $L$ which extend continuously to $X$. It follows from the construction that for each $x \in Y \setminus L = X$, $\lim(x) = \{x\}$.
The content of the following proposition is that the Higson compactification behaves nicely with respect to saturations, thus making it useful for the study of asymptotic properties of leaves.

**Proposition 6.32.** Let $X$ be a foliated space endowed with a continuous path metric on the leaves, amking them complete (for example, if $X$ is compact). If $F$ is a leaf of $X$ and $e$ is a point in the Higson corona of $F$, them $\lim(e)$ is a saturated subset of $X$.

*Proof.* Let $x \in X$ be a point in the limit set $\lim(e)$ of the point $e \in F^\nu$. As shown in Proposition 6.21, if $U$ is a neighborhood of $e$ in $\nu F$, then the open set $U \cap F$ contains metric balls of $F$ of arbitrarily large radius. Thus, for any such such neighborhood $U$ of $e$ there is a sequence $x_n \in U \cap F$ such that $x_n \to \infty$ in $F$, $x_n \to x$ in $X$, and a sequence of radii $r_n$ increasing to $\infty$ such that the metric balls $B(x_n, r_n) \subset U \cap F$. Hence, there exists a foliation chart $V$ in $X$ surrounding $x$ and so small that if $V$ meets one of the balls $B(x_n, r_n)$, then the whole plaque of $V$ is contained in that ball. This implies that the plaque of $x$ in $V$ is contained in $\bar{U} \cap \bar{F}$. The conclusion follows from the definition of $\lim(e)$. $\square$

**Definition 6.33.** A leaf $F$ of a foliated space $X$ is said to be *Higson recurrent* if the limit set of each point of the Higson corona of $F$ is dense in $X$.

Cantwell and Conlon [5] introduced the concept of totally recurrent leaf to denote a leaf each of whose ends is dense in the ambient space. They showed that the set of totally recurrent leaves of a foliated space, if non-empty, is residual. The Higson recurrence behaves in the following manner.

**Proposition 6.34.** Let $X$ be a compact foliated space, endowed with a metric on the leaves. Then $X$ is minimal if and only if $X$ has a Higson recurrent leaf.

*Proof.* Suppose that $F$ is a Higson recurrent leaf, but that there exists a non-empty, closed saturated set $Y$ which is a proper subset of $X$. The leaf $F$ is dense (because it is Higson recurrent) and so it approaches $Y$. For each $r > 0$ let $B(r)$ be a ball of radius $r$ in a leaf of $Y$. A normal neighborhood of $B(r)$ must then meet $F$ in an open set $B'(r)$ which contains a ball of radius $r'$, for any number $r' > 0$ as close to $r$ as desired. Therefore, there is a sequence of balls $B_n$ in $F$ of radius increasing to $\infty$ and such that $\bigcap_m \text{Cl}_X(\cup_{n \ge m} B_n) \subset Y$. Because of Proposition 6.21, the open sets $\cup_{n \ge m} B_n = U_n \cap F$, for some open neighborhood $U$ of a point $e$ in the Higson corona of $F$, thus contradicting that $F$ is Higson recurrent.

Suppose now that $X$ is minimal. Let $F$ be a leaf of $X$, which is dense. The limit set of a point of the Higson corona is saturated and closed. It is also non-empty because $X$ is compact. Let $x$ be a point in a leaf $F'$ which is approached by $F$. By the hypothesis on bounded geometry, the leaf $F'$ contains balls about $x$ of arbitrarily large radius. These balls lift to a neighborhood of a point $e$ of the Higson corona of $F$ ***, thus showing that $F'$ is in the limit set of $e$. $\square$

The following is the complement to Proposition 4.29.
**Theorem 6.35.** Let $X$ be a minimal, compact foliated space. If $B$ is a borel subset of $X$, then there exists a residual saturated subset $Y$ of $X$ such that, for each leaf $L \subset Y$, the intersection $F \cap B$ approaches all the points of the Higson corona of $L$.

*Proof.* It was shown in Proposition 4.29 that there is a set $Y$ as required such that the intersection $F \cap Y$ is a net in $F$. The conclusion follows from the description of the neighborhoods of points in the Higson corona of the leaf $F$ given in Proposition 6.21. □

*** *** ***

A space $X$ is said to be *weakly homogeneous* when, for all $x, y \in X$, every neighbourhood of $x$ contains an open subset homeomorphic to some neighbourhood of $y$. Two spaces $X$ and $Y$ are said to be *weakly homogeneously equivalent* if, for all $x \in X$ and $y \in Y$, every neighbourhood of $x$ contains an open subset homeomorphic to some neighbourhood of $y$ and viceversa. Note that weakly homogeneously equivalent spaces have the same topological dimension.

**Theorem 6.36.** For any minimal compact foliated space, the Higson corona of any leaf without holonomy is weakly homogoneus, and the Higson coronas of all leaves without holonomy are weakly homogoneusly equivalent; in particular, they have the same topological dimension.

*Proof.***

<figure><img src="image_12.png" /></figure>

## 6.5 Asymptotic dimension of leaves

*** *** *** This preliminary material on the properties of the Higson compactification was needed to deal with one asymptotic invariant, the first one appearing in [14], called the asymptotic dimension.

**Definition 6.37.** The asymptotic dimension as $\dim_+ M$ of a proper metric space $M$ with distance $d$ is $\le n$ if and only if for any number $R > 0$ there exists uniformly bounded families $\mathcal{M}_1, \dots, \mathcal{M}_{n+1}$ of subsets of $M$ such that $M = \bigcup_{i=1}^{n+1} \mathcal{M}_i$ is a cover of $M$, and such that for every $i = 1, \dots, n+1$ all the pairwise distances between members of $\mathcal{M}_i$ are $> R$.

The relevance of the as $\dim_+$ invariant of a metric space is illustrated by the following recent theorem of Dranishnikov, Keesling and Uspenkij [8].

**Theorem 6.38.** Let $M$ be a proper metric space. Then the dimension $\dim M^h$ of the Higson corona does not exceed the asymptotic dimension as $\dim_+ M$.

Two examples illustrate the invariant as $\dim_+$. For the hyperbolic space $\mathbb{H}^n$, the upper bound as $\dim_+ \mathbb{H}^n = n$ is claimed in [14]. For euclidean space $\mathbb{R}^n$, it is proven in [8] that as $\dim_+ \mathbb{R}^n = n$.

The main result of this section is the following theorem.
**Theorem 6.39.** Let $X$ be a compact minimal foliated space. Then there is a saturated residual subset $N$ of $X$ such that as $\dim_+ L$ is the same for every leaf $L$ in $N$.

With the asymptotic dimension being a quasi-isometry invariant, it suffices to consider the discrete version, as it simplifies the writing. That is, as it has already been done before, the leaves are replaced by the orbits of a finitely generated group $F$ acting, without holonomy, on a polish space $Z$.

For each integer $n > 0$ and real numbers $R, D > 0$, let $Y(R, D, n)$ be the union of orbits $O$ which admit families of subsets $\mathcal{V}_0, \dots, \mathcal{V}_n$ such that

(1) the diameter of each element of $\mathcal{V}_i$ is $\le D$;

(2) the distance between each pair of points of $\mathcal{V}_i$ is $\ge R$; and

(3) the union of all the elements of $\mathcal{V}_i$ covers $O$.

The union of orbits with asymptotic dimension $\le n$ is the set

$$ \bigcap \bigcup_{R \ D} Y(R, D, n). $$

For $r > 0$, let $Y(r, R, D, n)$ be the set of elements $x \in Z$ for which there exists families $\mathcal{V}_0, \dots, \mathcal{V}_n$ of subsets of the ball $B(x, r)$ in the orbit of $x$ satisfying the above conditions (1), (2), and (3) (replacing $O$ by $B(x, R)$). Note that we have the following relations:

If $R < R'$, then $Y(r, R, D, n) \supset Y(r, R', D, n)$, and if $D < D'$, then $Y(r, R, D, n) \supset Y(r, R', D, n)$. Here the number $r$ is either positive, or else $r = \infty$, in which case $Y(\infty, R, D, n) = Y(R, D, n)$.

**Lemma 6.40.** Each $Y(r, R, D, n)$ is a closed subset of $Z$.

*Proof.* Let $(x_k)$ be a sequence in $Y(r, R, D, n)$ which converges to a point $x \in Z$. Since there is no holonomy, the balls $B(x, r)$ are quasi-isometric to those $B(x_k, r)$ for $k$ sufficiently large. Therefore $x$ also has the credentials required to belong to $Y(r, R, D, n)$. $\square$

**Lemma 6.41.** The following identity holds:

$$ Y(R, D, n) = \bigcap_{r} Y(r, R, D, n). $$

*Proof.* For an orbit $O$, a point $x \in O$, and $r > 0$, there exists a natural restriction $\mathcal{V} \mapsto \mathcal{V}|_{B(x,r)}$, which assigns to a family $\mathcal{V}$ of subsets of $O$ the intersections of the elements of $\mathcal{V}$ with $B(x, r)$. It is clear that if the family $\mathcal{V}_0, \dots, \mathcal{V}_n$ satisfies the above properties in $O$, then the restrictions $\mathcal{V}_0|_{B(x,r)}$ satisfy the same properties relative to $B(x, r)$. It follows that $Y(R, D, n) \subset Y(r, R, D, n)$ for all $r > 0$.

For $r < r'$, there is also a natural restriction assigning families of $B(x, r')$ to families of $B(x, r)$. It is then clear that

$$ Y(r, R, D, n) \supset Y(r', R, D, n). $$
A locally finite graph is constructed having vertices the elements $(r, \mathcal{V}_0, \dots, \mathcal{V}_n)$, where $r$ is a positive integer and the $\mathcal{V}_k$ are families of subsets of $B(x, r)$ as above. There is an edge from $(r, \mathcal{V}_0, \dots, \mathcal{V}_n)$ to $(r+1, \mathcal{U}_0, \dots, \mathcal{U}_n)$ if and only if $\mathcal{U}_k|_{B(x,R)} = \mathcal{V}_k$ for all $k$. As the number of families $\{\mathcal{V}_k\}$ which satisfy the conditions in $B(x, r)$ is finite, this graph is locally finite. On the other hand, the fact that $x$ belongs to $\bigcup_r Y(r, R, D, n)$ implies that this graph has arbitrarily large paths. Therefore, there is an infinite increasing sequence $r_i$, and for each $i$ there are families

$$ \mathcal{V}_{i,0}, \dots, \mathcal{V}_{i,n} $$

of subsets of $B(x, r_i)$ satisfying the three conditions, and such that, whenever $i < j$,

$$ \mathcal{V}_{i,k} = \mathcal{V}_{j,k}|_{B(x,r_j)} $$

for all $k = 0, \dots, n$. If $\mathcal{V}_k$ is defined as the family of subsets of $O$ which belong to all the elements of a tail of the sequence $\{\mathcal{V}_{i,k}\}_i$, then it is easy to verify that the families $\mathcal{V}_0, \dots, \mathcal{V}_n$ satisfy the required properties in $O$. Hence also $x \in Y(R, D, n)$, and the proof is complete. $\square$

**Corollary 6.42.** Each $Y(R, D, n)$ is closed in $Z$, and each $\bigcup_D Y(R, D, n)$ is a countable union of closed subsets of $Z$.

*Proof.* The first assertion follows from the two previous lemmas. The second follows from the first because the union $\bigcup_D Y(R, D, n)$ does not change if $D$ varies among natural numbers. $\square$

<figure><img src="image_13.png" /></figure>

There are now two cases to consider.

**Case 1.** There exists $n$ such that for all $R > 0$ the set $\bigcup_D Y(D, R, n)$ has nonempty interior.

In this situation, as the action being considered is minimal and each $Y(R, D, n)$ is saturated, it follows that $Z = \bigcup_D Y(R, D, n)$. In particular, all the orbits have as $\dim_+ \le n$.

If $n_0$ is the smallest integer $n$ for which $\bigcup_D Y(D, R, n)$ has nonempty interior, then all orbits have asymptotic dimension bounded by $n_0$. Therefore $\bigcup_D Y(D, R, n)$ is meager for all $n < n_0$ because of Corollary 6.42, and hence so is $\bigcap_R \bigcup_D Y(D, R, n)$. It follows that the union of all leaves having asymptotic dimension exactly $n_0$ is residual.

**Case 2.** For every finite $n$ there exists $R$ such that $\bigcup_D Y(R, D, n)$ has empty interior.

By Corollary 6.42, if $\bigcup_D Y(R_0, D, n)$ has empty interior for some $R_0$, then it is a meager set. Therefore so is $\bigcap_R \bigcup_D Y(D, R, n)$, i.e., the union of leaves with as $\dim_+ \le n$. Since this is true for all $n$, the union of leaves with finite asymptotic dimension is meager. Hence all the leaves in a residual saturated set have as $\dim_+ = \infty$.
# Chapter 7

## Generic cohomology

### 7.1 Coarse cohomology of the leaves

Coarse cohomology of spaces was intrduced in Roe [?]. A very quick definition was given by Schmidt [24], using the concept of Grothendieck topologies.

Let $X$ be a metric space. The coarse topology $X_{co}$ on $X$ is the Grothendieck topology associated to the following data:

* $\mathrm{Cat}(X_{co})$ is the category of all open subsets of $X$, with inclusions as morphisms.

* A family $\{U_i \hookrightarrow U\}_{i \in I}$ of morphisms in $\mathrm{Cat}(X_{co})$ is a covering if every bounded subset $V \subset U$ is contained in $U_i$ for some $i \in I$.

**Example 7.1.** Let $X$ be a metric space (complete, proper?). If $U \subset X$, and $U_i = U \cap B(x, i)$, $i = 1, 2, \dots$, then $U_i \to U$ is a covering of $X$.

Associated to $X_{co}$ are the categories of presheaves (contravariant functors from $\mathrm{Cat}(X_{co})$ into the category of abelian groups) and sheaves, that is, those presheaves $F$ for which the sequence

$$F(U) \to \prod_{i \in I} F(U_i) \Rightarrow \prod_{i,j \in I} F(U_i \times_U U_j)$$

is exact for every covering $\{U_i \to U\}_{i \in I}$.

Let $A$ be an abelian group. Then $A : U \to A(U) = A$ for all $U \subset X$ is a presheaf on $X$, called the constant sheaf to $A$. This is in fact a sheaf in the coarse topology $X_{co}$ because if $\{U_i \to U\}$ be a covering in $X_{co}$ and $U_i \cap U_j \neq \emptyset$, for some indices $i, j \in I$, then there is an $U_k$ which meets both $U_i$ and $U_j$.

Let $F$ be a sheaf on $X_{co}$ and $\sigma$ a section of $F$ over $X$. For every real number $r > 0$, the $r$-support of $\sigma$ is the subset of $X$ defined as the union of all subset $U$ of $X$ with $\mathrm{diam} U < r$ and $s|_U \neq 0$. The $r$-support of $\sigma$ is denoted by $\mathrm{supp}_r(\sigma)$.

The group of sections of $F$ with compact support is defined by

$$\Gamma_c(X_{co}; F) = \{\sigma \in \Gamma(X_{co}, F) \mid \mathrm{supp}_r(\sigma) \text{ is bounded for every } r > 0.\}.$$
Then $\Gamma_c(X_{co}; -)$ is an additive, left exact functor from the category of sheaves on $X_{co}$ to the category of abelian groups. The coarse cohomology groups of $X$ with values in the sheaf $F$ are defined as the right derived functors of $\Gamma_c(X_{co}; -)$,

$$H_c^k(X_{co}; F) = R^k \Gamma_c(X_{co}; -)(F).$$

Let $X, Y$ be metric spaces. A subset $R$ of $X \times Y$ is a uniformly bornologous and uniformly proper relation if every $r > 0$ there exists an $s > 0$ such that if $U$ is subset of $X$ or of $Y$ and has diameter $diam(U) < r$, then the set $R^{-1}U$ has diameter $diam(R^{-1}U) < s$. The relation $R$ is surjective if for every $x$ there is $y \in Y$ such that $xRy$, and viceversa.

A relation from $X$ to $Y$ is a map $f : 2^X \to 2^Y$ which is completely additive, that is $f(\bigcup U_i) = \bigcup f(U_i)$.

If $X$ is a metric space, then the diameter function $diam$ is well defined on $2^X$. (the diameter of the empty set is $-\infty$?)

**Example 7.2.** A map $f : X \to Y$ is uniformly bornologous if for every $R > 0$ there is $S > 0$ such that, for all $x, x'$ in $X$, if $d(x, x') < r$, then $d(f(x), f(x')) < s$.

A map $f : X \to Y$ is uniformly proper if for every $r > 0$ there is $s > 0$ such that for all $x, x'$ in $X$, if $d(f(x), f(x')) < r$, then $d(x, x') < s$.

Suppose that $f : X \to Y$ is uniformly bornologous, uniformly proper, and surjective. Define a relation $R$ on $X \times Y$ to be the relation implied by $f$, that is, $(x, y) \in R$ if $f(x) = y$. Then $R$ is surjective. If $U$ is a subset of $X$ of diameter $diam(U) < r$, then $f(U)$ has diameter $diam f(U) < s$.

If $R$ is a relation on $X \times Y$ then $R$ induces functors $R_* : \mathrm{Sh}(X_{co}) \to \mathrm{Sh}(Y_{co})$. If $F$ is a sheaf on $X$ and $V$ is a subset of $Y$, then $R_*F(V) = F(R^{-1}V)$.

It is elementary to show that if $R$ is uniform, then $R$ takes coverings of $Y$ to coverings of $X$, and takes coverings of $X$ to coverings of $Y$.

$R$ induces functors $R_* : \mathrm{Cat}(X_{co}) \to \mathrm{Cat}(Y_{co})$ and $R^* : \mathrm{Cat}(Y_{co}) \to \mathrm{Cat}(X_{co})$. If $U \subset X$, then $R_*(U) = \{y \in Y \mid xRy \text{ for some } x \in U\}$.

## 7.2 Uniformly finite homology of the leaves

***

## 7.3 Algebraic asymptotic invariants

Let $\mathrm{Top}_\sigma$ denote the category of $\sigma$-compact topological spaces and surjective continuous maps, let $\mathbb{A}$ be a category with limits, and let $F : \mathrm{Top}_\sigma \to \mathbb{A}$ be a functor which is continuous in the following sense: For each object $X$ in $\mathrm{Top}_\sigma$ and each increasing sequence $K_n$ of compact subspaces of $X$, $F(\bigcup_n K_n) = \lim_n F(K_n)$, where the limit is injective or projective according to whether $F$ is covariant or contravariant.
If $F$ is a continuous functor, there is an associated functor $F^\infty$, the limit of $F$ at infinity, which is defined as follows. If $X$ is an object in $\mathrm{Top}_\sigma$, then

$$F^\infty(X) = \lim_K F(X \setminus K),$$

and if $f: X \to Y$ is surjective, then

$$F^\infty(f) = \lim_K F(X \setminus K \to Y \setminus K),$$

where $K$ denotes the family of compact subsets of $X$.

**Theorem 7.3.** Let $F$ be a continuous functor on $\mathrm{Top}_\sigma$ with values in the category of vector spaces over a field. Let $X$ be a transitive foliated space with no holonomy. Then the map $\dim: X \to \mathbb{N} \cup \{\infty\}$ which assigns to $x \in X$ the dimension of $F^\infty(L_x)$ is Borel and constant on a residual set of leaves.

*Proof.* Assume that $F$ is covariant. If $x \in X$, let $B(x, r)$ denote the ball of radius $r$ in the leaf thru $x$. For positive integers $n$ and $i < j < k < l$ let $Y_n(i, j, k, l)$ denote the set of points $x \in X$ for which the map

$$F(\overline{B(x, k)} \setminus B(x, j) \to \overline{B(x, l)} \setminus B(x, i))$$

has image of dimension $\ge n$. It easily follows by the already familiar arguments using recurrence that the sets $Y_n(i, j, k, l)$ are closed in $X$.

For each leaf $L$ of $X$ and each point $x \in L$, it holds that

$$F^\infty(L) = \lim_{\leftarrow i} \lim_{\to k>i} F(\overline{B(x, k)} \setminus B(x, i)),$$

from what it follows that the set of points $x \in X$ where $\dim F^\infty(L_x) \ge n$ is

$$Y_n = \bigcup_i \bigcap_{j>i} \bigcap_{k>j} Y_n(i, j, k, l).$$

Hence the function $\dim: X \to \mathbb{N}^*$, which is constant along the leaves, has the property that $\dim^{-1}[n, \infty]$ is borel for each $n \in \mathbb{N}$, and so $\dim^{-1}\{n\}$ is a borel saturated subset of $X$ for each $n \in \mathbb{N}$. The transitivity hypothesis on $X$ in turn implies that the function $\dim$ is constant on a residual saturated subset of $X$. $\square$

A functor $F$ to which this considerations apply is the cohomology $F(-) = H_c^0(-, \mathbb{Z}_2)$. The limit functor is $H_c^0(-)$, which may be interpreted as the Boolean algebra of the space of ends. The main theorem of Cantwell and Conlon [5] (and Ghys [11], although this one in a measure theoretic context) asserts that (for foliated spaces without compact leaves) not only the dimension of $H_c^0(L)$ is constant for all the leaves $L$ of a residual saturated set, but moreover this dimension is either 0, 1, 2 or $\infty$, and in this last case the end space of all such leaves is a Cantor set.
The family of compact sets $K \subset L$ form a directed set under inclusion. To each $K$
associate the group $H^q(L, L \times K)$, and to each inclusion $K \subset K'$ associate the natural map
$H^q(L, L \times K) \to H^q(L, L \times K')$. Then the direct limit $\lim \to H^q(L, L \times K)$ is the cohomology
of $L$ with compact supports, denoted by $H_c^q(L)$. Each element $\sigma$ of $H_c^q(L)$ can be represented by a cochain which vanish on all the singular simplices in the complement of a compact set
$K_\sigma$.

*****************************
# Chapter 8

## Generic differentiable geometry of leaves

### 8.1 Differentiable quasi-isometries

A map between Riemannian manifolds, $\phi: M \to M'$, is said to be a *quasi-isometry* if it is $C^1$ and there is some $C \ge 1$ such that its tangent map $\phi_*: TM \to TM'$ satisfies

$$ (8.1) \qquad \frac{1}{C} |\xi| \le |\phi_*(\xi)| \le C |\xi| $$

for all $\xi \in TM$; any such a constant $C$ will be called a *distortion* of $\phi$, or more precisely a *distortion of order 1*; the best possible of such distortions is usually called “the” *distortion* of $\phi$.

To define higher order versions of quasi-isometries, we use the Sasaki metric on the tangent bundle $TM$ of a Riemannian manifold $M$ [?, ?], which is the Riemannian metric determined by the following conditions: the horizontal and vertical subbundles, $\mathcal{H}, \mathcal{V} \subset TTM$, are orthogonal to each other, and the canonical identities $\mathcal{H}_\xi \equiv T_xM \equiv \mathcal{V}_\xi$ are isometries for each $x \in M$ and $\xi \in T_xM$. Consider also the *unit tangent sphere bundle* $T^1M = \{\xi \in TM \mid |\xi| = 1\}$.

Quasi-isometries and distortions of arbitrary order can inductively defined as follows. First, the terms *quasi-isometries of order 1* and *distortions of order 1* are used for the quasi-isometries and distortions as defined above. Now suppose that quasi-isometries and distortions of order $m-1$ are defined for some integer $m \ge 2$. Then a map between Riemannian manifolds, $\phi: M \to M'$, is said to be a *quasi-isometry of order m* if it is $C^m$ and the map $\phi_*: T^1M \to TM'$ is a quasi-isometry of order $m-1$. In this case, any distortion of order $m-1$ of $\phi_*: T^1M \to TM'$ is called a *distortion of order m* of $\phi$; the best possible of such distortions of order $m$ is called “the” *distortion of order m* of $\phi$. If $\phi$ is a quasi-isometry of order $m$ for all $m \in \mathbb{Z}_+$, then $\phi$ is called a *quasi-isometry of order $\infty$*.

A collection $\Phi$ of maps between Riemannian manifolds is said to be a family of *equi-quasi-isometries of order m* for some $m \in \mathbb{Z}_+$ if it is a family of quasi-isometries of order $m$ with some common distortion of order $m$. In this case, any such a common distortion of order $m$
is called an *equi-distortion of order $m$ of $\Phi$; the best possible of such equi-distortions of order $m$ can be called “the” equi-distortion of order $m$ of $\Phi$. *** We may simply use the terms equi-quasi-isometries and equi-distortion when $m = 1$ ***. If $\Phi$ is a collection of equi-quasi-isometries of order $m$ for all $m \in \mathbb{Z}_+$, then it is called a family of equi-quasi-isometries of order $\infty$.

Let $m \in \mathbb{Z}_+ \cup \{\infty\}$. The inverse of a quasi-isometry of order $m$ is easily seen to be a quasi-isometry of order $m$, and two Riemannian manifolds are said to be quasi-isometric with order $m$ when there is a quasi-isometry of order $m$ between them. With more generality, the inverses of a family of equi-quasi-isometries of order $m$ form a family of equi-quasi-isometries of order $m$, and a collection of Riemannian manifolds is said to be equi-quasi-isometric with order $m$ when there are equi-quasi-isometries of order $m$ between all such Riemannian manifolds.

To clarify the concept of quasi-isometry of order 2, we need the following description of tangent map of order 2. Let $\phi : M \to M'$ be a $C^2$ map between Riemannian manifolds. Consider the splittings $TTM = \mathcal{H} \oplus \mathcal{V}$ and $TTM' = \mathcal{H}' \oplus \mathcal{V}'$, where $\mathcal{H}$ and $\mathcal{H}'$ are the horizontal subbundles, and $\mathcal{V}$ and $\mathcal{V}'$ are the vertical subbundles. Fix any $x \in M$ and $\xi \in TM$. We have the canonical identities

$$ (8.2) \qquad T_{\xi}TM = \mathcal{H}_{\xi} \oplus \mathcal{V}_{\xi} \equiv T_{x}M \oplus T_{x}M, $$

$$ (8.3) \qquad T_{\phi_*(\xi)}TM' = \mathcal{H}'_{\phi_*(\xi)} \oplus \mathcal{V}'_{\phi_*(\xi)} \equiv T_{\phi(x)}M' \oplus T_{\phi(x)}M'. $$

The pull-back Riemannian vector bundle $\phi^*TM'$ is endowed with the pull-back of the Riemannian connection of $M'$, and let $\phi_* : TM \to \phi^*TM'$ also denote the homomorphism over the identity of $M$ induced by $\phi$. Let $X$ be a $C^\infty$ local section of $TM$ around $x$ so that $X(x) = \xi$; thus $\phi_*X$ is a $C^1$ local section of $\phi^*TM'$ around $x$ satisfying $(\phi_*X)(x) = \phi_*\xi \in (\phi^*TM')_x \equiv T_{\phi(x)}M'$. Then, for any $\zeta \in TM$ and each $C^\infty$ function $f$ defined on some neighborhood of $x$, we have

$$ \begin{aligned} \nabla_{\zeta}(\phi_*(fX)) - \phi_*(\nabla_{\zeta}(fX)) &= f(x) \nabla_{\zeta}(\phi_*X) + df(\zeta) \phi_*\xi \\ &\quad - f(x) \phi_*(\nabla_{\zeta}X) - df(\zeta) \phi_*\xi \\ &= f(x) (\nabla_{\zeta}(\phi_*X) - \phi_*(\nabla_{\zeta}X)) \end{aligned} $$

in $(\phi^*TM')_x \equiv T_{\phi(x)}M'$. Therefore $\nabla_{\zeta}(\phi_*X) - \phi_*(\nabla_{\zeta}X)$ depends only on $\zeta$ and $\xi$, and thus will be denoted by $A_{\phi}(\zeta \otimes \xi)$; indeed, the assignment $\zeta \otimes \xi \mapsto A_{\phi}(\zeta \otimes \xi)$ defines a continuous section $A_{\phi}$ of $(TM \otimes TM)^* \otimes \phi^*TM'$. Observe that $X$ can be chosen so that $\nabla_{\zeta}X = 0$, yielding $A_{\phi}(\zeta \otimes \xi) = \nabla_{\zeta}(\phi_*X)$ in this case. Then, from the definitions of tangent map and covariant derivative, it easily follows that, according to (8.2) and (8.3),

$$ (8.4) \qquad \phi^{**}\xi(\zeta_1, \zeta_2) \equiv (\phi_*\zeta_1, \phi_*\zeta_2 + A_{\phi}(\zeta_1 \otimes \xi)) $$

for all $\zeta_1, \zeta_2 \in T_x M$. Notice that, by the presence of $A_{\phi}$ in (8.4), it would be too restrictive to use $TM$ instead of $T^1M$ in the definition of quasi-isometries of arbitrary order.

The following following terminology is used in the next result: a collection of sections of Riemannian vector bundles are said to be *equi-bounded* if all of them are bounded with a common bound, which is called an *equi-bound* of the family.
**Lemma 8.1.** The following conditions are equivalent for any family $\Phi$ of $C^2$ maps between Riemannian manifolds:

(i) $\Phi$ is a collection of equi-quasi-isometries of order 2.

(ii) $\Phi$ is a collection of equi-quasi-isometries of order 1 and the sections $A_\phi$ are equi-bounded for $\phi \in \Phi$.

(iii) There is some $C \ge 1$ such that map $\phi : M \to M'$ in $\Phi$ satisfies

$$ \frac{1}{C} |\zeta| \le |\phi_{**\xi}(\zeta)| \le C |\zeta| $$

for all $\xi \in T^1 M$ and every $\zeta \in T_\xi TM$

Furthermore, in this case, the equi-distortions of order 2 of $\Phi$ are the constants $C \ge 1$ satisfying property (iii), and all of them are equi-distortions of order 1 too.

*Proof.* Assume that $\Phi$ satisfies property (iii) with some constant $C \ge 1$. Then, by taking $\zeta \in T_\xi T^1 M \subset T_\xi TM$ in property (iii), we get property (i) and moreover $C$ is an equi-distortion of order 2 of $\Phi$.

Now, suppose that $\Phi$ satisfies property (i), and let $C \ge 1$ be an equi-distortion of order 2 of $\Phi$. Take any $\phi : M \to M'$ in $\Phi$, any $x \in M$ and any $\xi \in T_x^1 M$. Since the horizontal subspace of $T_\xi TM$ is contained in $T_\xi T^1 M$, it is enough to prove property (iii) for all vertical vectors in $T_\xi TM$. Any such a vector is of the form $(0, \zeta)$ for some $\zeta \in T_x M$ according to (8.2). Hence

$$ |\phi_{**\xi}(0, \zeta)| \le |\phi_{**\xi}(\zeta, 0)| \le C |(\zeta, 0)| = C |(0, \zeta)| $$

by (8.4). This shows that $\Phi$ satisfies property (iii) with the given constant $C$.

From (8.4), it easily follows that property (iii) implies property (ii), and that any equi-distortion of order 2 of $\Phi$ is an equi-distortion of order 1.

Finally, assume that $\Phi$ satisfies property (ii). Let $C \ge 1$ be an equi-distortion of order 1 of $\Phi$, and let $K \ge 0$ be an equi-bound of the sections $A_\phi$ for $\phi \in \Phi$. Fix any $\phi : M \to M'$ in $\Phi$, $x \in M$, $\xi \in T_x^1 M$ and $\zeta_1, \zeta_2 \in T_x M$. According to (8.4), we have

$$ \begin{aligned} |\phi_{**\xi}(\zeta_1, \zeta_2)| &\le |\phi_*\zeta_1| + |\phi_*\zeta_2| + A_\phi(\zeta_1 \otimes \xi)| \\ &\le C |\zeta_1| + C |\zeta_2| + K |\zeta_1| \\ &\le (C + K) (|\zeta_1| + |\zeta_2|) \\ &\le (C + K) \sqrt{2} |(\zeta_1, \zeta_2)|. \end{aligned} $$
Moreover, for $0 < \lambda < 1$ with $K\lambda < 1/C$, we get

$$
\begin{align*}
|\phi_{**\xi}(\zeta_1, \zeta_2)| &\ge \frac{1}{\sqrt{2}} (|\phi_*\zeta_1| + |\phi_*\zeta_2 + A_\phi(\zeta_1 \otimes \xi)|) \\
&\ge \frac{1}{\sqrt{2}} (|\phi_*\zeta_1| + \lambda |\phi_*\zeta_2 + A_\phi(\zeta_1 \otimes \xi)|) \\
&= \frac{1}{\sqrt{2}} (|\phi_*\zeta_1| + |\phi_*(\lambda\zeta_2) + A_\phi(\zeta_1 \otimes \lambda\xi)|) \\
&\ge \frac{1}{\sqrt{2}} (|\phi_*\zeta_1| + |\phi_*(\lambda\zeta_2)| - |A_\phi(\zeta_1 \otimes \lambda\xi)|) \\
&\ge \frac{1}{\sqrt{2}} \left( \left( \frac{1}{C} - K\lambda \right) |\zeta_1| + \frac{\lambda}{C} |\zeta_2| \right) \\
&\ge \frac{1}{C'} (|\zeta_1| + |\zeta_2|) \\
&\ge \frac{1}{C'} |(\zeta_1, \zeta_2)|
\end{align*}
$$

for any $C' \ge 1$ so that

$$
\frac{1}{C'} \le \frac{1}{\sqrt{2}} \min \left\{ \frac{1}{C} - K\lambda, \frac{\lambda}{C} \right\} .
$$

Therefore $\Phi$ satisfies property (iii). $\square$

<figure><img src="image_14.png" /></figure>

Of course, there is an obvious generalization of Lemma 8.1 for higher orders, but it will
not be used here.

Let $M$ and $M'$ be Riemannian manifolds. For every $m \in \mathbb{N} \cup \{\infty\}$, consider the weak $C^m$ topology on $C^m(M, M')$ [?].

**Lemma 8.2.** Assume that $M$ is connected, and that $M'$ is connected and complete. For $m \in \mathbb{Z}_+ \cup \{\infty\}$, let $\Phi$ be a family of equi-quasi-isometries of order $m$ of $M$ to $M'$. Then, given any point $x_0 \in M$, the set $\Phi$ is precompact in $C^{m-1}(M, M')$ if and only if $\Phi(x_0) = \{\phi(x_0) \mid \phi \in \Phi\}$ is bounded in $M'$.

Proof. The “only if” part is obvious.

For finite $m$, the “if” part is proved by induction on $m$. For $m = 1^{***}$, the assumption
that $\Phi$ is a family of equi-quasi-isometries implies that $\Phi$ is equi-lipschitz, and thus equi-
continuous. On the other hand, for any $x \in M$, the set $\Phi(x) = \{\phi(x) \mid \phi \in \Phi\}$ is contained
in the penumbra Pen($\Phi(x_0)$, $Cr$), where $C \ge 1$ is an equi-distortion of $\Phi$ and $r > 0$ is the
distance from $x$ to $x_0$. So $\Phi(x)$ is precompact in $M'$ because $\Phi(x_0)$ is bounded and $M'$ is
complete. Therefore $\Phi$ is precompact in $C(M, M')$ by Ascoli's theorem.

Now take $m > 1$ and assume that the result holds for the case $m - 1$. The map $C^{m-1}(M, M') \to C^{m-2}(T^1M, TM')$, $\phi \mapsto \phi_*|_{T^1M}$, is an embedding. So it is enough to prove that the image $\Phi_*$ of $\Phi$ is precompact in $C^{m-2}(T^1M, TM')$, which holds by the inductive hypothesis because $\Phi_*$ is a family of equi-quasi-isometries of order $m - 1$.
The “if” part for $m = \infty$ can be easily proved as follows. In this case, we have proved that $\Phi$ is precompact in $C^m(M, M')$ for every finite $m$. By the continuity of the inclusion maps $C^{m+1}(M, M') \hookrightarrow C^m(M, M')$, it follows that $\Phi$ has the same closure $\bar{\Phi}$ in $C^m(M, M')$ and $C^{m+1}(M, M')$, and the weak $C^m$ and $C^{m+1}$ topologies coincide on $\bar{\Phi}$. Therefore $\bar{\Phi}$ is the closure of $\Phi$ in $C^\infty(M, M')$ too, and the weak $C^\infty$ and $C^m$ topologies coincide on $\bar{\Phi}$ for any finite $m$; thus $\Phi$ is precompact in $C^\infty(M, M')$ as desired. $\square$

**Corollary 8.3.** With the assumptions of Lemma 8.2, suppose that $\Phi$ is a collection of iso-metric embeddings $M \to M'$. Then, given any point $x_0 \in M$, the set $\Phi$ is precompact in $C^\infty(M, M')$ if and only if $\Phi(x_0) = \{\phi(x_0) \mid \phi \in \Phi\}$ is bounded in $M'$.

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

Iso($M$) is a lie group ***

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## 8.2 Quasi-isometric leaves

In this section, it will be described a geometric condition on the leaves of a transitive compact foliated space, and it will be shown to be a necessary and sufficient one for almost all the leaves to be (coarse) quasi-isometric. The first part of the discussion will establish that the union of all the leaves which are quasi-isometric to a fixed model metric space is a borel set. For this it is more convenient to work with the discrete version of the foliated space, namely, that in which the leaves are replaced by the orbits of a finitely generated group acting on a polish space. The condition also works in the more restrictive case of quasi-isometry via diffeomorphism, as it will be clear from the arguments.

Suppose that a finitely generated group $F$ acts on a space $X$. The group $F$ may be assumed to be a finitely generated free group, and endowed with an integer valued word metric which is right-invariant. The action of $F$ on itself by left multiplication is by quasi-isometries, the distortion constants depending only on the length of the acting element.

The orbit $Fx$ of a point $x$ inherits a canonical quasi-isometry type by identifying $Fx$ with the quotient space $F/F_x$, where $F_x$ is the stabilizer of $x$. The induced action of $F$ on the orbit $F/F_x$ is also by quasi-isometries.

**Proposition 8.4.** Let $X$ be a polish space on which a finitely generated group $F$ acts without holonomy. Then the equivalence relation “$x \sim y$ if and only if the orbits $Fx$ and $Fy$ are quasi-isometric” has borel equivalence classes, and it is a quotient of the orbit equivalence of the action.

Let $(M, m_0)$ be a fixed pointed metric space, of the type of a graph. Let $Y$ denote the subset of $X$ consisting of those points $x \in X$ such that the homogeneous space $(Fx, x)$ is pointed quasi-isometric to $(M, m_0)$.

It $y = sx$ is a point in the orbit of $x$, then $F_y = sF_x s^{-1}$, and the action of $s$ on the left induces a quasi-isometry

$$ (Fx, x) \to (Fy, y). $$
Therefore the space $Y$ just defined is a union of orbits.

Let $C, K, R > 0$ be constants, and let $Y(C, K, R)$ be the set of points $x \in X$ which satisfy the following. There are $K$-nets $A$ of $Fx$ and $N$ of $M$ which contain $x$ and $m_0$ respectively, and an injection $f: A \cap B_{Fx}(x, R)$ which satisfies (3) with the given $C$.

The map $f$ satisfies a lipschitz condition that yields that $f(A \cap F_{Fx}(x, R)) \subset N \cap B_M(m_0, CR)$. Furthermore $Y(C, K, R') \subset Y(C, K, R)$ if $R < R'$.

These sets satisfy the following two properties.

**Proposition 8.5.** *The set $Y(C, K, R)$ is open.*

This is elementary because $B_{Fy}(y, R)$ is almost a copy of $B_{Fx}(x, R)$ if $y$ is near $x$.

**Proposition 8.6.** *The set $Y$ satisfies*

$$Y = \bigcup_{C,K} \bigcap_{R} Y(C, K, R).$$

*Proof.* It will first be shown that $\cap_R Y(C, K, R)$ is the subspace of points $x$ of $X$ which satisfy the following: there are $K$-nets $A$, $N$ of $Fx$, $M$, containing $x$ and $m_0$, respectively, and a bijection $f: A \to N$ with $f(x) = m_0$ which satisfies (3) with the given $C$.

By hypothesis, given $R$, the map $f_R$ takes $A_R \cap B_{Fx}(x, R)$ into $N_R$ because $x$ belongs to $Y(C, K, R)$, where $A_R, f_R$, and $N_R$ depend on $R$.

But $R, B_{Fx}(x, R)$ and $B_M(m_0, CR)$ are finite, hence there are finitely may maps $A \cap B_{Fx}(x, R) \to N$ which satisfy (3), where $A, N$ are $K$-nets of $Fx$ and $M$, respectively. A standard argument with locally finite trees implies that there is a sequence $R_i$ increasing to $\infty$, and where all $A_i$ and $N_i$ are equal to $A, N$, and where all maps $f_i: A_{R_i} \cap B_{Fx}(x, R_i) \to N_{R_i}$ are restrictions of a single $f: A \to N$. This $f$ is the required quasi-isometry.

<figure><img src="image_15.png" /></figure>

One also has the following, which is equivalent to the previous statement.

**Proposition 8.7.** *The union of orbits of a fixed quasi-isometry type is a countable union of closed sets.*

*Proof.* We let $Y$ be as above. Then $Y = \bigcup_{C,K} Y(C, K)$, and $Y(C, K)$ is the collection of points $x$ in $X$ for which there exist $K$-nets $A$ and $N$ in $Fx$ and $M$ respectively, and a bijection $f: A \to N$ with $f(x) = m_0$ satisfying (3). By the above, $Y = \bigcup_R Y(C, K, R)$; hence it is enough to show that $Y(C, K, R)$ is closed. To this end, let $(x_n)$ be a sequence in $Y(C, K, R)$ which converges to a point $x$ in $X$. If the orbit of $X$ has no holonomy, then for large $n$ the ball $B_{Fx_n}(x_n, R)$ is almost a copy of $B_{Fx}(x, R)$, from what it follows what was claimed.

<figure><img src="image_16.png" /></figure>

This discussion has the following consequence.

**Proposition 8.8.** *If some $Y(C, K)$ has non-empty interior, then $Y = X$ and all orbits are quasi-isometric. If all $Y(C, K)$ have empty interior, then the union of orbits of the same quasi-isometry type is meager, and there is an uncountable family of such types.*
The next step consists in showing that when some $Y(C, K)$ has non-empty interior, the leaves must have a very strong geometric property which is defined as follows.

**Definition 8.9.** A complete riemannian manifold $L$ is quasi-symmetric if there exists a point $x_0$ in $L$ and a constant $K > 0$ such that for each $x \in L$ and each $n \ge 0$ there is a quasi-isometry of distortion $\le K$ between $B(x, n)$ and $B(x_0, n)$.

A similar definition can be made for locally finite graphs. Examples of quasi-symmetric spaces include homogeneous spaces of Lie groups with compact isotropy groups, universal covers of compact manifolds, and finitely generated groups with the word metric. The reason for invoking this concept is the following theorem.

**Theorem 8.10.** Let $X$ be a transitive foliated space. If there is a residual set of quasi-isometric leaves, then there is also a residual set of quasi-symmetric leaves.

*Proof.* By Proposition 8.8, and preserving the same notation, if there is a residual set of quasi-isometric leaves, then there are constants $C$ and $K$ such that the set $Y(C, K)$ has non-empty interior. Let $U$ be an open set contained in $Y(C, K)$. Then, by Proposition 4.29, almost every leaf meets this open set in a uniform net. Thus, if $L$ is one of such leaves and if $x, y \in L \cap Y(C, K)$, then the metric spaces $(L, x)$ and $(L, y)$ are quasi-isometric to the model $(M, m_0)$ by quasi-isometries with dilatation constants at most $(C, K)$. By composition, $(L, x)$ and $(L, y)$ are quasi-isometric via a quasi-isometry of dilatation at most $(C^2, CK + K)$. $\square$

The next result shows that the converse is also true. The result is true both in the coarse and smooth categories.

**Theorem 8.11.** Let $X$ be a minimal foliated space. If it has a quasi-symmetric leaf $L$, then every leaf without holonomy is quasi-isometric to $L$.

The proof makes use of the following lemma, which is a quasi-isometry version of the Ascoli-Arzela theorem.

**Lemma 8.12.** Let $X$ and $Y$ be two reasonable complete metric spaces. Suppose that there is a sequence of quasi-isometries $f_n : B(x_0, R_n) \to Y$, where $x_0 \in X$ is fixed and $f_n(x_0) = y_0$. If $R_n \to \infty$, and there is an upper bound for the distortion of all $f_n$, then $X$ and $Y$ are quasi-isometric.

*Proof of 8.11.* Let $X$ be a leaf without holonomy and fix a point $x \in X$. Let $K > 1$ be also fixed. Given a number $R$, we may cover the ball $B(x, R)$ with flow boxes. Then there is a number $\varepsilon$ for which the following property holds. Let $T$ be a transversal thru $x$ of diameter $\varepsilon$ (with respect to a distance in $M$). Then there is a residual set $T' \subset T$, and a map $f : B(x, R) \times T' \to M$ which is a homeomorphism of foliations, and which has image in the subspace of $M$ consisting of leaves without holonomy. Moreover, the restriction of $f$ to each $B(x, R) \times \{t'\}$, $t' \in T'$, is a quasi-isometry of distortion at most $K$.

Note that here the meaning of quasi-isometry is that $f$ has differential $df$ of bounded norm, where in the domain we have the same metric in all the balls $B(x, R) \times \{t'\}$.
Let $L$ be a quasi-symmetric leaf, and let $x_0 \in L$ be as in the definition. If $L$ approaches $X$, then it should meet $T'$ in some point $y$, which means that there is a component $D(y)$ of the intersection $L \cap f(B(x, R) \times T')$ which contains $y$. Normal projection induces a quasi-isometry (of distortion $\le K$) $g: B(x, R) \to D(y)$. Hence its image $g(B(x, R))$ is contained in $B(y, RK)$ and contains $B(y, R/K)$. Now we compose $g$ with a quasi-isometry of $B(y, RK)$ to $B(y_0, KR)$ given by the defining property of $L$, after which we can apply Lemma 8.12. $\square$

A few consequences of the previous discussion are as follows.

**Corollary 8.13.** Suppose that $X$ is minimal compact foliated space which has a leaf $L$ quasi-isometric to a symmetric space $G/K$. Then there is a residual set of leaves quasi-isometric to $G/K$.

**Corollary 8.14.** Let $X$ be a minimal compact foliated with two dimensional leaves. Suppose that $X$ admits no invariant transverse measure and that there is a simply connected leaf. Then there is a residual set consisting of leaves quasi-isometric to the hyperbolic plane.

*Proof*. The tangent bundle to the leaves may be assumed to be oriented, otherwise a standard construction (which works as in manifolds) produces a double cover of $X$ with this property. There is a conformally equivalent metric, also continuous on $X$, for which all the leaves have constant curvature $-1$ (note that this metric will also descend to $X$ if it was constructed on the double cover, as the covering transformations would be conformal). Then almost all are hyperbolic discs, using the fact that one of them is. $\square$

**Corollary 8.15.** Let $G$ be a finitely generated commutative group acting minimally on a compact space $X$. Then there is a residual set of quasi-isometric leaves.

*Proof*. If $G$ is finitely generated and commutative, then any right-invariant word metric is also left invariant, from what it follows that $G$ acts by isometries on every one of its homogeneous spaces. Thus every orbit is a quasi-symmetric space. $\square$

**Remark.** It is tempting to conjecture that if all the orbits of a minimal action of a finitely generated group on a compact metric space have polynomial growth then there is a residual subset of quasi-isometric orbits. This is false***

**Question.** Which finitely generated groups $G$ have the property that all their homogeneous spaces are quasi-symmetric?

The final part of this section recalls a coarse invariant of manifolds, introduced in [3], which will give a rough idea of how common is for a foliated space to have a residual set of quasi-isometric leaves.

Let $(L,g)$ be a complete manifold of bounded geometry, and let $\varepsilon, R > 0$. An $(\varepsilon, R)$ quasi-tiling of $L$ is a collection $\{K_1, \cdots, K_d\}$ of compact metric spaces of diameter $\le R$ and a countable collection of $(1+\varepsilon, \varepsilon)$ quasi-isometries $f_i: K_{a_i} \to L$ such that every compact set $K \subset L$ is at hausdorff distance at most $\varepsilon$ from one of the $f_i(K_{a_i})$.

The following definition is from [3].
**Definition 8.16.** For $\varepsilon \ge 0$, the $\varepsilon$-complexity function of $L$ is the function $H(L, \varepsilon, R)$, defined to be the minimum number $d$ for which $L$ admits an $(\varepsilon, R)$ quasi-tiling with $d$ pieces. If no such $(\varepsilon, R)$ quasi-tiling exists, then $H(L, \varepsilon, R) = \infty$.

The manifold $L$ is said to have *bounded complexity* if there is $\varepsilon > 0$ such that $H(L, \varepsilon, R)$ is bounded as a function of $R$.

For fixed $R$, $H(L, \varepsilon, R)$ is a decreasing function of $\varepsilon$.

**Theorem 8.17.** Let $X$ be a compact minimal (transitive) foliated space. If there is a residual set of leaves which are quasi-isometric, then there is a residual set of leaves of bounded complexity. Conversely, if $L$ is a leaf without holonomy and of bounded complexity, then every leaf without holonomy is quasi-isometric to $L$.

*Proof.* The proof in the transitive case requires some minor modifications which are left to the reader.

If $L$ is quasi-symmetric in the sense of Definition 8.9, and $\varepsilon$ is sufficiently large, then the function $R \mapsto H(L, \varepsilon, R)$ is bounded. Indeed, it is sufficient to take the ball of radius $R$ as one piece of the quasi-tiling.

Suppose now that $L$ is a leaf without holonomy and of bounded complexity. For each $R > 0$, let $K_1(R), \dots, K_d(R)$ be a finite set of compact spaces as in the definition of quasi-tiling, where $d \ge H(L, \varepsilon, R)$.

Let $N$ be another leaf without holonomy, and let $x \in N$. For $R > 0$, the ball $B_N(x, R)$ can be lifted via a quasi-isometry of dilatation $(A, B) \le (1 + \varepsilon, \varepsilon)$ to a domain in the leaf $L$. We only need to wait long enough for $L$ to pass sufficiently close to $N$ along the ball $B_N(x, R)$.

The argument shows that there are finitely many quasi-isometry types of leaves in $X$. By the results of ??, there is a residual saturated set consisting of quasi-isometric leaves. $\square$

The geometric entropy of a manifold $L$ is the number

$$h(L) = \lim_{\varepsilon \to \infty} \limsup_{R \to 0} \frac{\log H(L, \varepsilon, R)}{R}.$$

This notion is from [3], where it is shown that if $L$ is a leaf of a compact foliated manifold, then the geometric entropy of $L$ is bounded above by the geometric entropy of the foliation (as defined by [12]). The next section discusses a class of foliated spaces whose holonomy pseudogroups have entropy zero (in the sense of [12]). In general, it is not true that if the entropy of a foliated space is zero, then there is a residual set of quasi-isometric leaves.
# Chapter 9

## A canonical Riemannian foliated space

It is well known that a connected Riemannian manifold is quasi-isometric to a leaf of some compact smooth foliated space if and only if it is complete and of bounded geometry. Here, the “only if” part is elementary, and the “if” part follows by using the Gromov space of pointed proper metric spaces. In this paper, it is proved that, indeed, the above conditions describe the family of Riemannian manifolds which are isometric to some leaf of some compact Riemannian foliated space. This is proved by introducing the generalized Gromov space of pointed proper metric spaces with a distinguished closed subset. Other generalized Gromov spaces can be defined by adding other structures to metric spaces. All of them provide a systematic method to construct compact foliated spaces by using a candidate for a leaf. In this construction, minimal foliated spaces correspond to quasi-repetitive leaves, and quasi-effective equicontinuous minimal foliated spaces correspond to quasi-periodic leaves. This method also shows that a compact transitive smooth foliated space with trivial holonomy groups is determined by any dense leaf endowed with enough structure inherited from the ambient space.

### 9.1 The Gromov space

Let $M$ be a metric space with distance function $d$. For any subset $A \subset M$ and any $r > 0$, the *penumbra* of $A$ of *radius* $r$ is the set

$$ \mathrm{Pen}(X, r) = \{x \in M \mid d(x, X) < r\}. $$

Then the *Hausdorff distance* between subsets $A, B \subset M$ is

$$ d_H(A, B) = \inf\{r > 0 \mid A \subset \mathrm{Pen}(B, r), B \subset \mathrm{Pen}(A, r)\}. $$
With the Hausdorff distance, the family of closed subsets of $M$ becomes a metric equivalence relation, where two closed subsets are related when their Hausdorff distance is finite. For instance, an equivalence class consists of the single empty set; thus the empty set is an isolated point of the corresponding topological space. If $M$ is compact, then all non-empty subsets of $M$ are related and form a compact metric space with the Hausdorff distance.

In [13], M. Gromov used the Hausdorff distance to define a distance between any two abstract metric spaces $M$ and $N$. For that purpose, a metric on $M \cup N$ is said to be admissible if it restricts to the given metrics on $M$ and $N$. Then the gromov-Hausdorff distance between $M$ and $N$ is

$$d_{GH}(M, N) = \inf_{d} d_{H}(M, N),$$

where $d$ runs over all admissible metrics on $M \cup N$. With the gromov-Hausdorff distance, the collection $M_c$ of isometry classes of (non-empty) compact metric spaces becomes a metric space. Moreover $M_c$ is separable and complete [13, 21]; notice that a dense countable subset of $M$ is defined by the finite metric spaces with rational valued distance functions.

The meaning of the gromov-Hausdorff distance is clarified by the following result, whose proof is easy.

**Lemma 9.1.** We have $d_{GH}(M, N) < R$ if and only if there are maps $\phi: M \to N$ and $\psi: N \to M$ such that $d(\phi(x), \phi(x')) < d(x, x') + 2R$, $d(\psi(y), \psi(y')) < d(y, y') + 2R$, $d(\psi\phi(x), x) < 2R$ and $d(\phi\psi(y), y) < 2R$ for all $x, y \in M$ and all $x', y' \in N$.

From Lemma 9.1 it easily follows that, as indicated in [13], two metric spaces $M$ and $N$ are large scale Lipschitz equivalent (or, equivalently, coarsely quasi-isometric) if and only if there are bi-Lipschitz bijections $M \to M'$ and $N \to N'$ for some metric spaces $M'$ and $N'$ with $d_{GH}(M', N') < \infty$.

With the gromov-Hausdorff distance, the whole family $M$ of isometry classes of metric spaces becomes a metric equivalence relation, where two isometry classes of metric spaces are related just when their gromov-Hausdorff distance is finite. Lemma 9.1 shows how easily the gromov-Hausdorff distance can be infinite when non-compact metric spaces are considered, so the induced topology on $M$ is not very usefull.

A more interesting space, involving all reasonable non-compact metric spaces, can be defined by considering the gromov-Hausdorff distance between apropriate compact subsets. Such subsets will be closed balls centered at a distinguished point; thus we must consider pointed proper metric spaces. First, for pointed metric spaces $(M, x)$ and $(N, y)$, their pointed gromov-Hausdorff distance is

$$d_{GH}((M, x), (N, y)) = \inf_{d}\{d_{H}(M, N) + d(x, y)\},$$

where $d$ runs over all admissible metrics on $M \cup N$. As before, with the pointed gromov-Hausdorff distance, the family of isometry classes of pointed compact metric spaces becomes a complete separable metric space, denoted by $M_{c,*}$. Now, the Gromov space $M_*$ is the collection of isometry classes $[M, x]$ of pointed proper metric spaces $(M, x)$, endowed with
the gromov topology, which is defined by the following condition. The convergence $[M_i, x_i] \to [M, x]$ in $\mathcal{M}_*$ means that $d_{GH}((\bar{B}(x_i, r), x_i), (\bar{B}(x, r), x)) \to 0$ as $i \to \infty$ for each $r > 0$.

**Lemma 9.2.** $\mathcal{M}_*$ is a polish space.

*Proof.* A compatible distance $D$ on $\mathcal{M}_*$ can be defined by setting

$$D([M, x], [N, y]) = \sum_{n=1}^{\infty} \frac{1}{2^n} \min \{1, d_{GH}((\bar{B}(x, n), x), (\bar{B}(y, n), y))\}.$$

The completeness of $\mathcal{M}_*$ with $D$ follows easily from the completeness of $\mathcal{M}_{c,*}$ with $d_{GH}$. On the other hand, a countable dense subset of $\mathcal{M}_*$ is defined by the finite pointed metric spaces with rational valued distance functions. $\square$

For each proper metric space $M$, there is a canonical map $\iota_M : M \to \mathcal{M}_*$ defined by $\iota_M(x) = [M, x]$, whose continuity is easy to check; this map will be simply denoted by $\iota$ when there is no ambiguity. All such sets $\iota(M)$ form a canonical partition of $\mathcal{M}_*$. Each class $\mathcal{C}$ of proper metric spaces defines the saturated subset

$$\iota(\mathcal{C}) = \bigcup_{M \in \mathcal{C}} \iota(M) \subset \mathcal{M}_*.$$

M. Gromov has given simple characterizations of the classes of proper metric spaces defining precompact subsets of $\mathcal{M}_*$ [13], [21]. As a consequence, for any $n \in \mathbb{N}$ and $c \in \mathbb{R}$, the family of all complete connected Riemannian manifolds of dimension $n$ whose ricci curvature is $\ge c$ defines a precompact subset of $\mathcal{M}_*$, but this set is not closed in $\mathcal{M}_*$ for $n \ge 2$.

For a metric space $M$, let $\text{Iso}(M)$ denote the group of isometries of $M$. The canonical map $\iota : M \to \mathcal{M}_*$ induces an injective map $\bar{\iota}_M : \text{Iso}(M)\backslash M \to \mathcal{M}_*$, which may be simply denoted by $\bar{\iota}$ if no confusion can arise. The following terminology will be used:

* $M$ is said to be *aperiodic* when $\text{Iso}(M) = \{\text{id}_M\}$, and $M$ is said to be *periodic* otherwise.

* $M$ is said to be *locally aperiodic* when each point $x \in M$ has a neighborhood $U_x$ such that

$$\{h \in \text{Iso}(M) \mid h(x) \in U_x\} = \{\text{id}_M\},$$

and $M$ is said to be *locally periodic* otherwise.

Observe that the existence of such $U_x$ is equivalent to the existence of a neighbourhood $V_x$ such that

$$\{h \in \text{Iso}(M) \mid V_x \cap h(V_x)\} = \{\text{id}_M\}.$$

The following result is obvious.

**Lemma 9.3.** For any proper metric space $M$, the map $\iota : M \to \mathcal{M}_*$ is (locally) injective if and only if $M$ is (locally) aperiodic.

***
## 9.2 Convergence of Riemannian manifolds

Fix a compact domain<fnref n="1" /> $\Omega \subset \mathbb{R}^n$ and any $m \in \mathbb{N}$. The $C^m$ norm on the vector space $C^m(\Omega, \mathbb{R}^k)$ of $C^m$ functions $\Omega \to \mathbb{R}^m$ is defined by

$$ \|f\|_{C^m} = \sum_{|I| \le m} \sup_{x \in \Omega} |\partial_I f(x)|, $$

where $\partial_I f = \partial^{|I|} f / \partial x^I$ for any multi-index $I$—the usual multi-index notation is used with the standard coordinates $(x^1, \dots, x^n)$—. Then $C^m(\Omega, \mathbb{R}^k)$ becomes a banach space with the $C^m$ norm.

Hölder spaces are also needed for the appropriate statement of the convergence results that will be used later. For $0 < \alpha \le 1$, the $C^\alpha$-pseudonorm of any function $f: \Omega \to \mathbb{R}^k$ is

$$ \|f\|_{\alpha} = \sup_{x,y \in \Omega} \frac{|f(x) - f(y)|}{|x - y|^{\alpha}}. $$

When $\alpha = 1$, this is just the best Lipschitz constant for $f$. If $f$ is $C^m$, its $C^{m,\alpha}$ norm is

$$ \|f\|_{C^{m,\alpha}} = \|f\|_{C^m} + \sum_{|I|=m} \|\partial_I f\|_{\alpha}. $$

The notation $\|f\|_{C^m, \Omega}$ and $\|f\|_{C^{m,\alpha}, \Omega}$ can be also used to specify the domain, and the notation $C^\alpha$ can be used instead of $C^{0,\alpha}$.

The Hölder space $C^{m,\alpha}(\Omega, \mathbb{R}^k)$ consists of $C^m$ functions $\Omega \to \mathbb{R}^k$ with finite $C^{m,\alpha}$ norm; this is a banach space with the $C^{m,\alpha}$ norm. The $C^{m,\alpha}$ norm can be defined as well when $\Omega$ is not compact, but the space of functions with finite $C^{m,\alpha}$ norm is too small in this case. Instead, when $\Omega$ is not compact, the Hölder space $C^{m,\alpha}(\Omega, \mathbb{R}^k)$ consists of the functions $\Omega \to \mathbb{R}^k$ with $C^{m,\alpha}$ restrictions to any compact subdomain of $\Omega$, which is a fréchet space with the topology defined by the condition that the convergence of a sequence in $C^{m,\alpha}(\Omega, \mathbb{R}^k)$ means the $C^{m,\alpha}$ convergence of the corresponding sequence of restrictions to any compact subdomain of $\Omega$. Similarly, when $\Omega$ is not compact, the fréchet space $C^m(\Omega, \mathbb{R}^k)$ can be defined by using $C^m$ convergence instead of $C^{m,\alpha}$ convergence.

An important property of Hölder spaces is that, if $0 < \beta < \alpha \le 1$ and $\Omega$ is compact, then the closed bounded subsets of $C^{m,\alpha}(\Omega, \mathbb{R}^k)$ are compact in $C^{m,\beta}(\Omega, \mathbb{R}^k)$ (see e.g. [21]).

Another relevant property is the following. Let $\Omega \subset \mathbb{R}^n$ be a bounded open domain of diameter $\le D$, let $K \subset \Omega_0$ be a compact subdomain, and let $\delta > 0$ be the distance between $K$ and the complement of $\Omega$. For smooth functions $a^{i,j}$ and $b^i$ on $\Omega$ with $a^{i,j} = a^{j,i}$, the differential operator $L = a^{i,j} \partial_i \partial_j + b^i \partial_i$ is elliptic if $(a^{i,j})$ is positive definite at every point. Assume the stronger property that there is some $\lambda > 1$ such that the eigenvalues of $(a^{i,j})$ at each point lie in $[1/\lambda, \lambda]$ and $\|a^{i,j}\|_{C^{m,\alpha}}, \|b^i\|_{C^{m,\alpha}} \le \lambda$ for some $m \ge 0$. Then the Schauder estimates (or elliptic estimates) state that, for $0 < \alpha \le 1$, there is some $C > 0$, depending

<footnote n="1">¹All domains considered in this paper will be assumed to have smooth boundary.</footnote>
on $n, m, \alpha, \lambda, \delta$ and $D$, such that the following inequalities hold for any smooth function $f$ on $\Omega$:

$$ \begin{aligned} \|f\|_{C^{m+2,\alpha},K} &\le C (\|Lf\|_{C^{m,\alpha},\Omega} + \|f\|_{C^{\alpha},\Omega}), \\ \|f\|_{C^{1,\alpha},K} &\le C (\|Lf\|_{C^0,\Omega} + \|f\|_{C^{\alpha},\Omega}). \end{aligned} $$

If furthermore $\Omega$ has smooth boundary, then there is some $C > 0$, depending on $n, m, \alpha, \lambda$ and $D$, such that

$$ \|f\|_{C^{m+2,\alpha},\Omega} \le C (\|Lf\|_{C^{m,\alpha},\Omega} + \|f\|_{C^{m+2,\alpha},\partial\Omega}) $$

for every function $f$ smooth on $\bar{\Omega}$.

By using charts in the standard way, we can define a category of $C^{0,1}$ manifolds and $C^{0,1}$ maps, which turns out to be just the category of Lipschitz manifolds and locally Lipschitz maps. Nevertheless, the composite of a $C^\alpha$ map and a $C^\beta$ map is $C^{\alpha\beta}$ for $0 < \alpha, \beta \le 1$, and thus a category of $C^\alpha$ manifolds and $C^\alpha$ maps can not be defined for $0 < \alpha < 1$. But $C^{m,\alpha}$ tensors over any smooth manifold $M$ can be defined by requiring them to have $C^{m,\alpha}$ local coefficients with respect to any chart of the smooth structure of $M$. Moreover the $C^{m,\alpha}$ convergence of a sequence of $C^{m,\alpha}$ tensors means the $C^{m,\alpha}$ convergence of the corresponding local coefficients with respect to any chart of the smooth structure of $M$; this condition defines the $C^{m,\alpha}$ topology on the space of $C^{m,\alpha}$ tensors over $M$, which becomes a fréchet space. When $M$ is compact, the space of $C^{m,\alpha}$ tensors over $M$ can be endowed with a banach space structure by considering the norm defined in a standard way by using a finite atlas of $M$, a subordinated partition of unity and the $C^{m,\alpha}$ norm of compactly supported $C^{m,\alpha}$ tensors on $\mathbb{R}^n$. The $C^m$ topology on the space of $C^m$ tensors can be defined similarly by using just $C^m$ convergence of the local coefficients; this gives rise to a fréchet space, which again can be endowed with a banach space structure if $M$ is compact by using a finite atlas and a subordinated partition of unity. If $M$ is a compact Riemannian manifold, then an equivalent norm on the space of $C^m$ tensors $T$ over $M$ is defined by

$$ \|T\|_{C^m} = \sum_{k=0}^{m} \sup_{x \in M} |\nabla^k T(x)|, $$

which can be called the $C^m$ norm. If this norm is considered on some compact domain $\Omega$ of a Riemannian manifold, then the notation $\|T\|_{C^m,\Omega}$ may be used. The limit of the $C^m$ topologies on the space of $C^\infty$ tensors over a manifold $M$ is called the $C^\infty$ topology; again, this gives rise to a fréchet space; in particular, if $M$ is a compact Riemannian manifold, then a complete metric $d_\infty$ inducing this topology is given by

$$ (9.1) \qquad d_\infty(T, T') = \sum_{m=0}^{\infty} \frac{1}{2^m} \min \{1, \|T - T'\|_{C^m}\}. $$

Of course, similar definitions can be given for manifolds with boundary and sections of any smooth vector bundle.
For the sake of simplicity, let us introduce the following notation. For a Riemannian manifold $M$, its metric tensor can be denoted by $g_M$, or simply $g$ if there is no ambiguity. For a family of Riemannian manifolds $M_i$ with $i$ running in some index set, the notation $g_i = g_{M_i}$ will be used. The analogous notation will be used for the corresponding distance functions (when the manifolds are connected), $d_M$, $d$ or $d_i$; for the curvature tensor, $\mathcal{K}_M$, $\mathcal{K}$ or $\mathcal{K}_i$; for the ricci curvature, $\mathrm{Ric}_M$, $\mathrm{Ric}$ or $\mathrm{Ric}_i$; and for the injectivity radius, $\mathrm{inj}_M$, $\mathrm{inj}$ or $\mathrm{inj}_i$.

For each $n \in \mathbb{N}$, let $\mathcal{M}_*(n)$ denote the set of isometry classes of pointed complete connected Riemannian manifolds of dimension $n$. Like for $\mathcal{M}_*$, the isometry class of each pointed Riemannian manifold $(M, x)$ will be denoted by $[M, x]$. For $m \in \mathbb{N}$ and $0 < \alpha \le 1$, the $C^{m,\alpha}$ topology on $\mathcal{M}_*(n)$ is defined by requiring that the convergence $[M_i, x_i] \to [M, x]$ means that, for each compact domain $\Omega \subset M$, there are $C^\infty$ embeddings $\phi_i : \Omega \to M_i$ for large enough $i$ such that $\phi_i(x) = x_i$ and $\phi_i^* g_i \to g$ on $\Omega$ with respect to the $C^{m,\alpha}$ topology. The corresponding space will be denoted by $\mathcal{M}_*^{m,\alpha}(n)$. If $C^m$ or $C^\infty$ convergence is used instead of $C^{m,\alpha}$ convergence, we get the $C^m$ or $C^\infty$ topology on $\mathcal{M}_*(n)$, and the corresponding space will be denoted by $\mathcal{M}_*^m(n)$ or $\mathcal{M}_*^\infty(n)$, respectively. The notation $\mathcal{M}_*^\alpha(n)$ can be also used instead of $\mathcal{M}_*^{0,\alpha}(n)$. Notice that the $C^{m,\alpha}$ topology on $\mathcal{M}_*(n)$ is finer than the $C^m$ topology and coarser than the $C^{m+1}$ topology. Moreover the $C^{m,\alpha}$ topology gets finer as $m$ or $\alpha$ increases, and $\mathcal{M}_*^\infty(n)$ is the inverse limit of the spaces $\mathcal{M}_*^{m,\alpha}(n)$ as $m \to \infty$; i.e., a set is open in $\mathcal{M}_*^\infty(n)$ if and only if it is open in some $\mathcal{M}_*^{m,\alpha}(n)$.

Since any complete connected Riemannian manifold is a proper metric space by the theorem of Hopf-Rinow, there is a canonical map $\mathcal{M}_*(n) \to \mathcal{M}_*$, which is obviously injective and with saturated image; $\mathcal{M}_*(n)$ will be considered as a subset of $\mathcal{M}_*$ in this sense. This inclusion map is continuous with respect to the $C^{m,\alpha}$ or $C^\infty$ topology on $\mathcal{M}_*^\infty(n)$.

**Lemma 9.4.** Each space $\mathcal{M}_*^\infty(n)$ is completely metrizable.

*Proof.* For $m, r \in \mathbb{Z}_+$, a pseudo-metric $D_{m,r}$ on the set $\mathcal{M}_*(n)$ can be defined by

$$D_{m,r}([M_1, x_1], [M_2, x_2]) = \max \left\{ \inf_{\phi} \|g_1 - \phi^* g_2\|_{C^{m,\Omega_1}}, \inf_{\psi} \|g_2 - \psi^* g_1\|_{C^{m,\Omega_2}} \right\},$$

where $\phi$ and $\psi$ run over the $C^\infty$ embeddings $\Omega_1 \to M_2$ and $\Omega_2 \to M_1$ for arbitrary compact domains $\Omega_i \subset M_i$ containing the balls $B(x_i, r)$ and such that $\phi(x_1) = x_2$ and $\psi(x_2) = x_1$. Then a distance function inducing the topology of $\mathcal{M}_*^\infty(n)$ can be defined by

$$D_\infty([M_1, x_1], [M_2, x_2]) = \sum_{m,r=1}^{\infty} \frac{1}{2^{m+r}} \min\{1, D_{m,r}([M_1, x_1], [M_2, x_2])\}.$$

The completeness of $D_\infty$ follows easily from the completeness of the space of $C^\infty$ tensors over a compact manifold with the metric (9.1). $\square$

Of course, an analogous argument shows that each space $\mathcal{M}_*^{m,\alpha}(n)$ is polish, but it will not be used here.
For each complete connected Riemannian manifold $M$ of dimension $n$, the canonical map $\iota: M \to \mathcal{M}_*$ defines continuous maps $\iota: M \to \mathcal{M}_*^{m,\alpha}(n)$ and $\iota: M \to \mathcal{M}_*^{\infty}(n)$, and the injection $\bar{\iota}: \text{Iso}(M)\setminus M \to \mathcal{M}_*$ defines continuous injections $\bar{\iota}: \text{Iso}(M)\setminus M \to \mathcal{M}_*^{m,\alpha}(n)$ and $\bar{\iota}: \text{Iso}(M)\setminus M \to \mathcal{M}_*^{\infty}(n)$. The proof of the following result is an easy exercise involving only the definition of the space $\mathcal{M}_*^{\infty}(n)$.

**Lemma 9.5.** *The following properties hold:*

(i) Consider the $C^\infty$ topology on $\iota(M)$. Then $\iota: M \to \iota(M)$ is closed if and only if, for any $x \in M$, any sequence of compact domains $\Omega_i$ in $M$ and any sequence of $C^\infty$ embeddings $\phi_i: \Omega_i \to M$ satisfying $\|g - \phi_i^*g\|_{C^m, \Omega_i} \to 0$ as $i \to \infty$ for any $m \in \mathbb{N}$, the sequence $\phi_i(x)$ is bounded in $M$.

(ii) The map $\bar{\iota}: \text{Iso}(M)\setminus M \to \mathcal{M}_*^{\infty}(n)$ is an embedding if and only if, for any $x \in M$, any sequence of compact domains $\Omega_i$ in $M$ and any sequence of $C^\infty$ embeddings $\phi_i: \Omega_i \to M$ so that $d(x, \phi_i(x)) \to \infty$ and $\|g - \phi_i^*g\|_{C^m, \Omega_i} \to 0$ as $i \to \infty$ for any $m \in \mathbb{N}$, there is some sequence of isometries $\psi_i: M \to M$ such that

$$ \sup\{d(\phi_i(y), \psi_i(y)) \mid y \in B(x, r)\} \to 0 $$

as $i \to \infty$ for some $r > 0$.

## 9.3 Compact families of Riemannian manifolds

The following “norm” of Riemannian manifolds was introduced by Petersen to clarify some results about convergence of Riemannian manifolds [21]. Let $m \in \mathbb{N}$, $0 < \alpha \le 1$ and $r > 0$, and let $A$ be a subset of a Riemannian manifold $M$. The $C^{m,\alpha}$ norm on the scale $r$ of $A \subset M$, denoted by $\|A \subset M\|_{C^{m,\alpha},r}$, is the infimum of all $Q > 0$ such that there is a family of charts of $M$, $\phi_s = (x_s^1, \dots, x_s^n): U_s \to \mathbb{R}^n$, with $\phi_s(U_s) = B(0, r)$ and satisfying the following conditions:

(n1) Every ball $B(x, \frac{1}{10} e^{-Q}r)$ in $M$ with $x \in A$ is contained in some $U_s$.

(n2) $e^{-Q} \le |\phi_{s*}| \le e^Q$ on $U_s$.

(n3) $r^{|I|+\alpha} \|\partial^{|I|}g_{s; i, j}/\partial x_s^I\|_{\alpha} \le Q$ for all multi-indices $I$ with $0 \le |I| \le m$, where the functions $g_{s; i, j}$ on $B(0, r)$ are the local coefficients of the metric tensor $g$ on $U_s$.

(n4) $\|\phi_t \circ \phi_s^{-1}\|_{C^{m+1,\alpha}} \le (10 + r) e^Q$.

The notations $\|M\|_{C^{m,\alpha},r}$ or $\|x \in M\|_{C^{m,\alpha},r}$ are respectively used when $A = M$ or $A = \{x\}$. The $C^m$ norm on the scale $r$ of $A \subset M$ can be similarly defined by using the $C^0$ norm is used in (n3), and the notation $\|A \subset M\|_{C^{m,\alpha},r}$, $\|M\|_{C^{m,\alpha},r}$ or $\|x \in M\|_{C^{m,\alpha},r}$ is used in this case. The following result is proved in [21].

**Proposition 9.6.** If $A \subset M$ is precompact, then the following properties hold:
(i) $\|A \subset (M, g)\|_{C^{m,\alpha},r} = \|A \subset (M, \lambda^2 g)\|_{C^{m,\alpha},\lambda r}$ for all $\lambda > 0$.

(ii) The function $r \mapsto \|A \subset M\|_{C^{m,\alpha},r}$ is continuous and converges to 0 as $r \to 0$.

(iii) If $[M_i, x_i] \to [M, x]$ in $\mathcal{M}_*^{m,\alpha}(n)$, then there are precompact domains $A_i \subset M_i$ so that $\|A_i \subset M_i\|_{C^{m,\alpha},r} \to \|A \subset M\|_{C^{m,\alpha},r}$ for all $r > 0$. When all the manifolds are closed, we can let $A = M$ and $A_i = M_i$.

For $m, n \in \mathbb{N}$, $0 < \alpha \le 1$ and $Q, r > 0$, let $\mathcal{M}_*^{m,\alpha}(n, Q, r)$ denote the subset of $\mathcal{M}_*(n)$ defined by the class of complete connected Riemannian $n$-manifolds $M$ with $\|M\|_{C^{m,\alpha},r} \le Q$. The set $\mathcal{M}_*(n, Q, r)$ can be defined similarly by using the $C^m$ norm on the scale $r$. The following important theorem is essentially due to Cheeger [7, 21].

**Theorem 9.7** (Fundamental Theorem of Convergence Theory). Given $m, n \in \mathbb{N}$, $0 < \beta < \alpha \le 1$ and $Q, r > 0$, the set $\mathcal{M}_*^{m,\alpha}(n, Q, r)$ is compact in $\mathcal{M}_*^{m,\beta}(n)$.

The proof of this theorem is in two main steps. First, the set $\mathcal{M}_*^{m,\alpha}(n, Q, r)$ is shown to be precompact in the $\mathcal{M}_*$. Second, limit points in $\mathcal{M}_*$ of sequences in $\mathcal{M}_*^{m,\alpha}(n, Q, r)$ are shown to belong to $\mathcal{M}^{m,\beta}(n, Q, r)$, and in fact the convergence is in $\mathcal{M}_*^{m,\beta}(n)$.

Several important results about convergence of manifolds can be proved by using Theorem 9.7. The proofs of the sharpest ones also use *harmonic coordinates*; that is, coordinate systems consisting of harmonic functions, which exists around each point in any Riemannian manifold [21].

Let $M$ be a Riemannian manifold with ricci curvature $Ric_M$ (or simply $Ric$). When harmonic coordinates are considered, the laplacian has the expression

$$ (9.2) \qquad \Delta = g^{i,j} \partial_i \partial_j $$

with the usual notation, and moreover the metric coefficients and the ricci curvature have the following relation:

$$ (9.3) \qquad \frac{1}{2} \Delta g + Q(g, \partial g) = - \text{Ric}, $$

where $g \equiv (g_{i,j})$, $\partial g = (\partial_1 g, \dots, \partial_n g)$ and $\text{Ric} \equiv (\text{Ric}_{i,j})$ on $U \equiv \phi(U)$, and $Q$ is some universal analytic expression that is polynomial in $g$, quadratic on $\partial g$ and has a denominator term depending on $\sqrt{\det g}$ [21].

Now, for $m \in \mathbb{N}$, $0 < \alpha \le 1$, $r > 0$ and a subset $A$ of $M$, the $C^{m,\alpha}$-harmonic norm on the scale $r$ of $A \subset M$, denoted by $\|A \subset M\|_{C^{m,\alpha},r}^{\text{harm}}$, can be defined by using the conditions (n1), (n2) and (n3) of the definition of $\|A \subset M\|_{C^{m,\alpha},r}$, and replacing condition (n4) by the requirement that all coordinate systems $\phi_s$ be harmonic. The notations $\|M\|_{C^{m,\alpha},r}^{\text{harm}}$ or $\|x \in M\|_{C^{m,\alpha},r}^{\text{harm}}$ can also be used when $A = M$ or $A = \{x\}$, respectively. From the Schauder estimates it easily follows that, for $0 < r_1 < r_2$, there is some $C > 0$, depending only on $n$, $m$, $\alpha$, $r_1$ and $r_2$, such that

$$ (9.4) \qquad \|M\|_{C^{m,\alpha},r_1} \le C \|M\|_{C^{m,\alpha},r_2}^{\text{harm}}. $$

Then the following harmonic analog of the fundamental theorem follows by showing that harmonic charts converge to a harmonic chart in any convergent sequence in $\mathcal{M}_*^{m,\alpha}(n)$ [2, 21].
**Corollary 9.8.** Given $m, n \in \mathbb{N}$, $0 < \beta < \alpha \le 1$ and $Q, r > 0$, the class of complete connected Riemannian $n$-manifolds $M$ with $\|M\|_{C^{m,\alpha},r}^{\text{harm}} \le Q$ defines a closed subset of $\mathcal{M}_*^{m,\alpha}(n)$ and a compact subset of $\mathcal{M}_*^{m,\beta}(n)$.

The following result is due to M.T. Anderson [2, 21].

**Proposition 9.9.** If $A \subset M$ is precompact, then the following properties hold:

(i) $\|A \subset (M, g)\|_{C^{m,\alpha},r}^{\text{harm}} = \|A \subset (M, \lambda^2 g)\|_{C^{m,\alpha},\lambda r}^{\text{harm}}$ for all $\lambda > 0$.

(ii) The function $r \mapsto \|A \subset M\|_{C^{m,\alpha},r}^{\text{harm}}$ is continuous. Moreover, when $m \ge 1$, it converges to 0 as $r \to 0$.

(iii) If $[M_i, x_i] \to [M, x]$ in $\mathcal{M}_*^{m,\alpha}(n)$ with $m \ge 1$, then there are precompact domains $A_i \subset M_i$ so that $\|A_i \subset M_i\|_{C^{m,\alpha},r}^{\text{harm}} \to \|A \subset M\|_{C^{m,\alpha},r}^{\text{harm}}$ for all $r > 0$. When all the manifolds are closed, we can let $A = M$ and $A_i = M_i$.

(iv) $\|A \subset M\|_{C^{m,\alpha},r}^{\text{harm}} = \sup_{x \in A} \|x \in M\|_{C^{m,\alpha},r}^{\text{harm}}$.

**Lemma 9.10.** Let $m, n \in \mathbb{Z}_+$, $0 < \alpha < 1$, $0 < r_1 < r_2$ and $\Lambda, K > 0$. There is some $C \ge 0$, depending on $m, n, \alpha, K, r_1, r_2$ and $\Lambda$, such that, if $|\nabla^k \text{Ric}| \le \Lambda$ for $0 \le k \le m-1$ and $\|A \subset M\|_{C^{m,\alpha},r_2}^{\text{harm}} \le K$, then $\|M\|_{C^{m,\alpha},r_1}^{\text{harm}} \le C$.

*Proof.* This result is proved by induction on $m$. The case $m=1$ was proved by M.T. Anderson [2, 21]. Suppose now that $m \ge 2$ and the result holds for $m-1$. Given any harmonic coordinate system satisfying the conditions to get $\|A \subset M\|_{C^{m,r_2}}^{\text{harm}} \le K$, we just have to find an appropriate bound for the $C^{m,\alpha}$ norm on $B(0, r_1)$ of the corresponding matrix $(g_{i,j}) \equiv g$ of local coefficients of the metric tensor. Let $r = (r_1+r_2)/2$. By (9.2), the laplacian satisfies the necessary conditions to apply the Schauder estimates. Hence

$$ \|g\|_{C^{m,\alpha},B(0,r_1)} \le C_1 (\|\Delta g\|_{C^{m-2,\alpha},B(0,r)} + \|g\|_{C^{\alpha},B(0,r)}) $$

for some $C_1 > 0$ depending only on $m, n, \alpha, K, r_1$ and $r$. On the other hand, from (9.3) and by using the expression of the christoffel symbols in terms of the metric coefficients, it follows that

$$ \|\Delta g\|_{C^{m-2,\alpha},B(0,r)} \le C_2 (\|g\|_{C^0,B(0,r)} + \|g\|_{C^{m-1,\alpha},B(0,r)}) $$

for some $C_2 > 0$ depending on $m, n, \alpha, K, r$ and $\Lambda$. So

$$ \|g\|_{C^{m,\alpha},B(0,r_1)} \le C_1(2C_2+1) \|g\|_{C^{m-1,\alpha},B(0,r)} $$

By the induction hypothesis, there is some $C_3 > 0$, depending on $m, n, \alpha, K, r, r_2$ and $\Lambda$, such that $\|g\|_{C^{m-1,\alpha},B(0,r)} \le C_3$, and the result follows with $C = C_1(2C_2+1)C_3$. $\square$

**Theorem 9.11.** Given $m, n \in \mathbb{N}$, $0 < \alpha < 1$ and $\Lambda, i_0, Q > 0$, there is some $r > 0$, depending on $m, n, \alpha, K, \Lambda, i_0$ and $Q$, such that, if a complete Riemannian $n$-manifold $M$ satisfies $\text{inj} \ge i_0$ and $|\nabla^k \text{Ric}| \le \Lambda$ for $0 \le k \le m-1$, then $\|M\|_{C^{m,\alpha},r}^{\text{harm}} \le Q$.
Proof. For $m = 1$, the proof is due to M.T. Anderson [2, 21]. A key part of Anderson's proof is the following assertion.

**Claim 15.** If a convergent sequence $[M_i, x_i] \to [M, x]$ in $M_*^{1,\alpha}(n)$ satisfies $\text{inj}_i \to \infty$, $|\text{Ric}_i| \to 0$ and $\|M_i\|_{C^{1,\alpha},1}^{\text{harm}} \le Q$, then $M$ is isometric to the euclidean space $\mathbb{R}^n$

The general case is just an adaptation of Anderson's proof by using higher order derivatives. Suppose that the statement is wrong; thus there is some $Q > 0$ such that, for each $i \in \mathbb{Z}_+$, there is a Riemannian manifold $M_i$ satisfying $\text{inj}_i \ge i_0$, $|\nabla^k \text{Ric}_i| \le \Lambda$ for $0 \le k \le m-1$ and $\|M_i\|_{C^{m,\alpha},1/i}^{\text{harm}} > Q$. By Proposition 9.9 (ii), there is some $r_i \in (0, 1/i)$ for each $i$ such that $\|M_i\|_{C^{m,\alpha},r_i}^{\text{harm}} = Q$. Let $N_i$ denote the manifold $M_i$ with the rescaled metric $h_i = r_i^{-2}g_i$, which satisfies $\text{inj}_{N_i} \ge i_0/r_i$, $|\nabla^k \text{Ric}_{N_i}| \le r_i^2 \Lambda$ for $0 \le k \le m$ and $\|N_i\|_{C^{m,\alpha},1}^{\text{harm}} = Q$ by Proposition 9.9 (i). So $\|x_i \in N_i\|_{C^{m,\alpha},1}^{\text{harm}} \ge Q/2$ for some $x_i \in N_i$ by Proposition 9.9 (iv). By Lemma 9.10, the sequence $\|N_i\|_{C^{m,\gamma},1}^{\text{harm}}$ is bounded for any $\gamma \in (\alpha, 1)$. Therefore, by Corollary 9.8, we can assume that the sequence $[N_i, x_i]$ converges in $M_*^{m,\alpha}(n)$ to some element $[N, x]$. Thus a contradiction follows because, on the one hand, we have $\|x \in N\|_{C^{m,\alpha},1}^{\text{harm}} \ge Q/2$ by Proposition 9.9 (iii), and on the other hand, $N$ is isometric to the euclidean space $\mathbb{R}^n$ by Claim 15. $\square$

For $m \in \mathbb{N}$, recall that a complete Riemannian manifold $M$ is said to be of bounded geometry of order $m$ when

$$ \inf_{x \in M} \text{inj}(x) > 0, \quad \sup_{x \in M} |\nabla^k \mathcal{K}(x)| < \infty $$

for all $k \in \{0, \dots, m\}$. If $M$ is of bounded geometry of any finite order, then it is said to be of bounded geometry of order $\infty$. A class $\mathcal{C}$ of complete Riemannian manifolds is said to be of equi-bounded geometry of order $m$ when

$$ \inf\{\text{inj}_M(x) \mid M \in \mathcal{C}, x \in M\} > 0, \quad \sup\{|\nabla^k \mathcal{K}_M(x)| \mid M \in \mathcal{C}, x \in M\} < \infty $$

for all $k \in \{0, \dots, m\}$. The class $\mathcal{C}$ is said to be of equi-bounded geometry of order $\infty$ if it is of equi-bounded geometry of any finite order. For the sake of simplicity, let us similarly define (equi-)ricci-bounded geometry of order $m \in \mathbb{N} \cup \{\infty\}$ by using the ricci curvature instead of the curvature tensor. Of course, (equi-)bounded geometry of order $m$ imply (equi-)ricci-bounded geometry of order $m$. The converse is almost true according to the following results.

**Corollary 9.12.** Let $m, n \in \mathbb{N}$. The following properties hold for any class $\mathcal{C}$ of complete connected Riemannian $n$-manifolds:

(i) If $\iota(\mathcal{C})$ is precompact in $M_*^{m+2}(n)$, then $\mathcal{C}$ is of equi-bounded geometry of order $m$.

(ii) If $\mathcal{C}$ is of equi-ricci-bounded geometry of order $m$, then $\iota(\mathcal{C})$ is precompact in $M_*^{m+1,\alpha}(n)$ for any $\alpha \in (0, 1)$.
*Proof.* Since the definition of the exponential map involves derivatives of order $\le 2$ of the metric coefficients, a lower semicontinuous map $\mathcal{M}_*^2(n) \to (0, \infty]$ is defined by $[M, x] \mapsto \mathrm{inj}_M(x)$. On the other hand, because the curvature involves derivatives of order $\le 2$ of the metric coefficients, a continuous map $\mathcal{M}_*^{k+2}(n) \to (0, \infty)$ is defined by $[M, x] \mapsto |\nabla^k \mathcal{K}_M(x)|$ for each $k \in \{0, \dots, m\}$. Then property (i) follows.

Property (ii) follows directly from (9.4) and Theorems 9.7 and 9.11. $\square$

<figure><img src="image_17.png" /></figure>

For $m, n \in \mathbb{N}$ and $0 < \alpha < 1$, let $\mathrm{Cl}_{\mathcal{M}_*}$, $\mathrm{Cl}_{m,\alpha}$ and $\mathrm{Cl}_{\infty}$ denote the closure operators on the spaces $\mathcal{M}_*$, $\mathcal{M}_*^{m,\alpha}(n)$ and $\mathcal{M}_*^{\infty}(n)$, respectively.

**Corollary 9.13.** The following properties are equivalent for a class $\mathcal{C}$ of complete connected Riemannian $n$-manifolds:

(i) $\mathcal{C}$ is of equi-bounded geometry of order $\infty$.

(ii) $\mathcal{C}$ is of equi-ricci-bounded geometry of order $\infty$.

(iii) The set $\iota(\mathcal{C})$ is precompact in $\mathcal{M}_*^{\infty}(n)$.

Moreover, in this case, we have $\mathrm{Cl}_{\mathcal{M}_*}(\iota(\mathcal{C})) = \mathrm{Cl}_{m,\alpha}(\iota(\mathcal{C})) = \mathrm{Cl}_{\infty}(\iota(\mathcal{C}))$ for all $m \in \mathbb{N}$ and $\alpha \in (0,1)$, and all of the spaces $\mathcal{M}_*$, $\mathcal{M}_*^{m,\alpha}(n)$ and $\mathcal{M}_*^{\infty}(n)$ induce the same topology on $\mathrm{Cl}_{\mathcal{M}_*}(\iota(\mathcal{C}))$.

*Proof.* Property (ii) is an obvious consequence of property (i), as we said.

Assume $\mathcal{C}$ satisfies property (ii). For $m \le m'$ in $\mathbb{N}$ and $0 < \alpha \le \alpha' < 1$, since the topology of $\mathcal{M}_*^{m',\alpha'}(n)$ is finer than the topology of $\mathcal{M}_*^{m,\alpha}(n)$, we get $\mathrm{Cl}_{m',\alpha'}(\iota(\mathcal{C})) \subset \mathrm{Cl}_{m,\alpha}(\iota(\mathcal{C}))$. Moreover $\mathrm{Cl}_{m',\alpha'}(\iota(\mathcal{C}))$ is compact in $\mathcal{M}_*^{m',\alpha'}(n)$ by Corollary 9.12, and thus it is compact in $\mathcal{M}_*^{m,\alpha}(n)$ as well. Because $\mathcal{M}_*^{m,\alpha}(n)$ is Hausdorff, it follows that $\mathrm{Cl}_{m',\alpha'}(\iota(\mathcal{C})) = \mathrm{Cl}_{m,\alpha}(\iota(\mathcal{C}))$, and furthermore the spaces $\mathcal{M}_*^{m',\alpha'}(n)$ and $\mathcal{M}_*^{m,\alpha}(n)$ induce the same topology on $\mathrm{Cl}_{m,\alpha}(\iota(\mathcal{C}))$. Since $\mathcal{M}_*^{\infty}(n)$ is the inverse limit of the identity maps $\mathcal{M}_*^{m',\alpha'}(n) \to \mathcal{M}_*^{m,\alpha}(n)$, we get $\mathrm{Cl}_{m,\alpha}(\iota(\mathcal{C})) = \mathrm{Cl}_{\infty}(\iota(\mathcal{C}))$, and the spaces $\mathcal{M}_*^{m,\alpha}(n)$ and $\mathcal{M}_*^{\infty}(n)$ induce the same topology on $\mathrm{Cl}_{\infty}(\iota(\mathcal{C}))$; in particular, $\iota(\mathcal{C})$ is precompact in $\mathcal{M}_*^{\infty}(n)$; i.e., $\mathcal{C}$ satisfies property (iii).

Since the canonical map $\mathcal{M}_*^{m,\alpha}(n) \to \mathcal{M}_*$ is compact and $\mathcal{M}_*$ is Hausdorff, we similarly get that $\mathrm{Cl}_{\mathcal{M}_*}(\iota(\mathcal{C})) = \mathrm{Cl}_{m,\alpha}(\iota(\mathcal{C}))$, and the spaces $\mathcal{M}_*$ and $\mathcal{M}_*^{m,\alpha}(n)$ induce the same topology on $\mathrm{Cl}_{\mathcal{M}_*}(\iota(\mathcal{C}))$, obtaining the last part of the statement.

Now assume that $\mathcal{C}$ satisfies property (iii). Hence $\iota(\mathcal{C})$ is precompact in each space $\mathcal{M}_*^{m,\alpha}(n)$ because its topology is coarser than the topology of $\mathcal{M}_*^{\infty}(n)$, and property (i) follows from Corollary 9.12. $\square$

<figure><img src="image_18.png" /></figure>

**Lemma 9.14.** Each space $\mathcal{M}_*^{\infty}(n)$ is polish.

*Proof.* By Lemma 9.4, it only remains to prove that $\mathcal{M}_*^{\infty}(n)$ is separable.

For each $a \in \mathbb{N}$, let $\mathcal{C}_a$ be the class of complete connected Riemannian manifolds $M$ such that $\mathrm{inj}_M(x) \ge 1/a$ and $|\nabla^k \mathcal{K}_M(x)| \le a$ for all $x \in M$ and all $k \in \mathbb{N}$, and consider the class $\mathcal{C} = \bigcup_{a=0}^{\infty} \mathcal{C}_a$. Since $\mathcal{M}_*$ is polish by Lemma 9.2, it also satisfies the second axiom of
countability. So its subspace $\iota(\mathcal{C}_a)$ is second countable as well. But the spaces $\mathcal{M}_*$ and $\mathcal{M}_*^{\infty}(n)$ induce the same topology on $\iota(\mathcal{C}_a)$ by Lemma 9.13. Hence $\iota(\mathcal{C}_a)$ is a second countable, and thus separable, as subspace of $\mathcal{M}_*^{\infty}(n)$. Therefore $\iota(\mathcal{C}) = \bigcup_{a=1}^{\infty} \iota(\mathcal{C}_a)$ is a separable subspace of $\mathcal{M}_*^{\infty}(n)$ too, and the result follows because $\iota(\mathcal{C})$ is easily seen to be dense in $\mathcal{M}_*^{\infty}(n)$. $\square$

## 9.4 A canonical Riemannian foliated space

Recall the definitions of aperiodicity and local aperiodicity from Section ??. Recall also that $\iota: M \to \mathcal{M}_*$ denotes the canonical map for each metric space $M$, which induces a canonical homeomorphism $\bar{\iota}: \text{Iso}(M)\setminus M \to \iota(M)$, and that the equivalence classes of $\mathcal{M}_*$ are the sets $\iota(M)$ (Section ??). For any $n \in \mathbb{N}$, let $\mathcal{M}_{*,a}(n)$ and $\mathcal{M}_{*,la}(n)$ be the saturated subsets of $\mathcal{M}_*(n)$ defined by complete connected Riemannian manifolds which are aperiodic and locally aperiodic, respectively. This makes sense only for $n \ge 2$ because $\mathcal{M}_{*,la}(1) = \emptyset^{***}$. The notation $\mathcal{M}_{*,a}^{\infty}(n)$ and $\mathcal{M}_{*,la}^{\infty}(n)$ will be used when these sets are endowed with the topology induced by $\mathcal{M}_*^{\infty}(n)$. The following is the main result of this section.

**Theorem 9.15.** The following properties hold for each integer $n \ge 2$:

(i) $\mathcal{M}_{*,la}^{\infty}(n)$ is open and dense in $\mathcal{M}_*^{\infty}(n)$.

(ii) $\mathcal{M}_{*,la}^{\infty}(n)$ with its natural equivalence relation is a foliated space *** of dimension $n$.

(iii) The foliated space $\mathcal{M}_{*,la}^{\infty}(n)$ has canonical $C^{\infty}$ and Riemannian structures such that, for each locally aperiodic complete connected Riemannian manifold $M$, the canonical homeomorphism $\bar{\iota}$ between $\text{Iso}(M)\setminus M$ and the leaf $\iota(M)$ is an isometry.

(iv) For each locally aperiodic complete connected Riemannian manifold $M$, the quotient map $M \to \text{Iso}(M)\setminus M$ corresponds to the holonomy covering of the leaf $\iota(M)$ by the canonical homeomorphism $\bar{\iota}: \text{Iso}(M)\setminus M \to \iota(M)$. In particular, the set $\mathcal{M}_{*,a}^{\infty}(n)$ is the union of leaves of $\mathcal{M}_{*,la}^{\infty}(n)$ with trivial holonomy groups, and thus $\mathcal{M}_{*,a}^{\infty}(n)$ is residual in $\mathcal{M}_{*,la}^{\infty}(n)$.

Before proving Theorem 9.15, let us indicate some of its consequences.

**Corollary 9.16.** For each integer $n \ge 2$, any saturated polish subspace of $\mathcal{M}_{*,la}^{\infty}(n)$ is a Riemannian foliated subspace of dimension $n$.

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

**Corollary 9.17.** Let $X$ be a compact saturated subset of $\mathcal{M}_*(n)$ for some integer $n \ge 2$. Then $X_{la} = X \cap \mathcal{M}_{*,la}(n)$ is a locally compact Riemannian foliated subspace of dimension $n$ of $\mathcal{M}_{*,la}^{\infty}(n)$.

*Proof.* By Corollary 9.13, the topologies of $\mathcal{M}_*$ and $\mathcal{M}_*^{\infty}(n)$ have the same restriction to $X$. So $X_{la}$ is open and dense in $X$ by Theorem 9.15-(i). Hence $X_{la}$ is a locally compact polish space by Lemma 9.2. Moreover $X_{la}$ is saturated in $\mathcal{M}_*$, and then the result follows from Corollary 9.16. $\square$
Let $X$ be a Riemannian foliated space. For each leaf $L$ of $X$, let $\tilde{L}$ denote its holonomy
cover with the induced Riemannian metric, and for each $x \in X$, let $L_x$ be the leaf of $X$ that
contains $x$ $**$. If the leaves of $X$ are complete, then a canonical map $\iota: X \to \mathcal{M}_*^\infty(n)$ is well
defined by $\iota(x) = [\tilde{L}_x, \tilde{x}]$, which is continuous by the local reeb stability theorem.

**Definition 9.18.** The Riemannian foliated space $X$ is said to be *holonomy-aperiodic* when, for all leaves $L$ and $M$ of $X$, if there is an isometry $\phi : \tilde{L} \to \tilde{M}$, then $L = M$ and $\phi$ is a deck transformation of the covering map $\tilde{L} \to L$.

Observe that the leaves of a holonomy-aperiodic Riemannian foliated space are aperiodic.
According to Theorem 9.15, the Riemannian foliated space $\mathcal{M}_{*,\text{la}}^{\infty}(n)$ is holonomy-aperiodic.
Indeed, the following shows certain partial universality of $\mathcal{M}_{*,\text{la}}^{\infty}(n)$ as holonomy aperiodic
Riemannian foliated space.

**Proposition 9.19.** Let $X$ be a compact Riemannian foliated space of dimension $n \ge 2$. Then $X$ is holonomy-aperiodic if and only if it is isometric to a Riemannian foliated subspace *** of $\mathcal{M}_{*,\text{la}}^{\infty}(n)$; more precisely, the canonical map $\iota: X \to \mathcal{M}_{*,\text{la}}^{\infty}(n)$ is an isometric embedding in this case.

*Proof*. The “if” part follows because holonomy aperiodicity is a hereditary property ***.

Now suppose that $X$ is holonomy-aperiodic to prove the “only if” part. Since $X$ is compact, its leaves are complete, and thus the canonical map $\iota: X \to \mathcal{M}_*^\infty(n)$ is well defined and continuous. Furthermore $\iota$ is injective with image in $\mathcal{M}_{*,\text{la}}^\infty(n)$ by the holonomy-aperiodicity of $X$. So $\iota: X \to \iota(X)$ is a homeomorphism by the compactness of $X$, and $\iota(X)$ is a Riemannian foliated subspace of dimension $n$ of $\mathcal{M}_{*,\text{la}}^\infty(n)$ by Corollary 9.17. Furthermore $\iota$ obviously preserves the foliated and Riemannian structures according to Theorem 9.15. $\square$

Let $X$ be a Riemannian foliated space of dimension $n \ge 2$. For $x, y \in X$, let $x \sim y$ mean
that $\tilde{L}_x$ is isometric to $\tilde{L}_y$. This defines an equivalence relation on $X$ which is compatible
with the foliated structure, and let $X_{\text{ha}}$ denote the corresponding quotient space endowed
with the equivalence relation induced by the foliated structure of $X$; this can be called the
maximal holonomy-aperiodic quotient of $X$.

**Proposition 9.20.** Suppose that $X$ is compact with locally aperiodic leaves. Then $X_{\text{ha}}$ is a foliated space which admits $C^{\infty}$ and Riemannian structures so that the quotient map $X \to X_{\text{ha}}$ is isometric, and moreover $X_{\text{ha}}$ is holonomy-aperiodic.

*Proof.* The map $\iota: X \to \mathcal{M}_{*,\text{la}}^{\infty}(n)$ induces a continuous injection $\bar{\iota}: X_{\text{ha}} \to \mathcal{M}_{*,\text{la}}^{\infty}(n)$, which indeed is an embedding by the compactness of $X_{\text{ha}}$. Moreover $\bar{\iota}$ preserves the equivalence relations. So, by Corollary 9.17, $X_{\text{ha}}$ is a foliated space, which can be endowed with $C^{\infty}$ and Riemannian structures so that $\bar{\iota}$ is an isometric embedding. Then the quotient map $X \to X_{\text{ha}}$ is isometric because so is $\iota$. $\square$

Proposition 9.19 can be slightly improved as follows to obtain a characterization of all Riemannian foliated subspaces of $\mathcal{M}_{*,\text{la}}^{\infty}(n)$.
**Definition 9.21.** Let $g$ denote the metric tensor of $X$, and let $\tilde{g}$ denote its lift to the holonomy cover of any leaf. The Riemannian foliated space $X$ is said to be strongly holonomy-aperiodic when, for points $x, x_i \in X$, with $i \in \mathbb{N}$, and for points $\tilde{x} \in \tilde{L}_x$ and $\tilde{x}_i \in \tilde{L}_{x_i}$ over $x$ and $x_i$, respectively, if for each compact domain $\Omega$ of $\tilde{L}$ containing $x$ there is a sequence of $C^\infty$ embeddings $\phi_i : \Omega \to \tilde{L}_{x_i}$ for $i$ large enough such that $\phi_i(x) = x_i$ and $\phi_i^* \tilde{g} \to \tilde{g}$ as $i \to \infty$ with respect to the $C^\infty$ topology, then $x_i \to x$ as $i \to \infty$.

It is easy to check that strong holonomy-aperiodicity implies holonomy-aperiodicity, and both of these properties are equivalent for compact Riemannian foliated spaces. Again, $M_{*,\text{la}}^\infty(n)$ is easily seen to be strongly holonomy-aperiodic; in fact, it is a universal strongly holonomy aperiodic Riemannian foliated space with complete leaves according to the following.

**Proposition 9.22.** Let $X$ be a Riemannian foliated space of dimension $n \ge 2$ whose leaves are complete. Then $X$ is strongly holonomy-aperiodic if and only if it is isometric to a Riemannian foliated subspace of $M_{*,\text{la}}^\infty(n)$; more precisely, the canonical map $\iota: X \to M_{*,\text{la}}^\infty(n)$ is an isometric embedding in this case.

*Proof.* The “if” part follows because strong holonomy-aperiodicity is a hereditary property too.

Suppose that $X$ is strongly holonomy-aperiodic to prove the “only if” part. Then $\iota: X \to M_{*,\text{la}}^\infty(n)$ is an isometric injection as in the proof of Proposition 9.19. So it is enough to prove that $\iota: X \to \iota(X)$ is closed, which is an easy consequence of the strong holonomy-aperiodicity. $\square$

Corollary 9.17 can be used to construct examples of compact Riemannian foliated spaces as follows. Let $M$ be a complete connected Riemannian manifold of dimension $n \ge 2$. Suppose that $M$ is of bounded geometry of order $\infty$. Then, by Corollary 9.13, the space $X = \text{Cl}_M(\iota(M))$ is compact and the topologies of $M_*$ and $M_*^\infty(n)$ have the same restriction to $X$. By Theorem 9.15 and Corollary 9.17, if all the equivalence classes in $X$ were defined by aperiodic Riemannian manifolds, then $X$ would be a compact riemanian foliated subspace of $M_{*,\text{la}}^\infty(n)$ whose leaves have trivial holonomy groups. This aperiodicity condition on the whole of $X$ can be described in terms of $M$ as follows.

**Definition 9.23.** A complete connected Riemannian manifold $M$ is said to be strongly aperiodic when, for any sequence of points $x_i \in M$, any sequence of compact domains $\Omega_i$ in $M$, and any sequence of $C^\infty$ embeddings $\phi_i : \Omega_i \to M$ such that

$$x_i \in \Omega_i, \quad \sup_i d(x_i, \phi_i(x_i)) < \infty, \quad d(x_i, \partial\Omega_i) \to \infty, \quad \|g - \phi_i^*g\|_{C^m, \Omega_i} \to 0$$

as $i \to \infty$ for any $m \in \mathbb{N}$, we have

$$\sup\{d(x, \phi_i(x)) \mid x \in \Omega_i \cap B(x_i, r)\} \to 0$$

as $i \to \infty$ for some $r > 0$. When this condition is satisfied for all sequences $x_i$ diverging to infinity$^2$, then $M$ is said to be *aperiodic at infinity*.

<footnote>$^2$This means that, for any bounded subset $B$ of $M$, there is some index $i_0$ such that $x_i \notin B$ for all $i \ge i_0$.</footnote>
**Remark.** The condition of Definition 9.23 for bounded sequences simply means the aperiodicity of $M$. So strong aperiodicity is equivalent to aperiodicity plus aperiodicity at infinity.

**Lemma 9.24.** The following properties any complete connected Riemannian manifold $M$ of dimension $n$:

(i) If $M$ is strongly aperiodic, then $\mathrm{Cl}_\infty(\iota(M)) \subset \mathcal{M}_{*,a}^\infty(n)$.

(ii) If $M$ is aperiodic at infinity, then $\mathrm{Cl}_\infty(\iota(M)) \setminus \iota(M) \subset \mathcal{M}_{*,a}^\infty(n)$.

(iii) Suppose that $M$ is of bounded geometry of order $\infty$. Then $M$ is strongly aperiodic if and only if $\mathrm{Cl}_\infty(\iota(M)) \subset \mathcal{M}_{*,a}^\infty(n)$.

(iv) Suppose that $M$ is of bounded geometry of order $\infty$ and $\iota: M \to \iota(M)$ is closed with respect to the $C^\infty$ topology on $\iota(M)$. Then $M$ is aperiodic at infinity if and only if $\mathrm{Cl}_\infty(\iota(M)) \setminus \iota(M) \subset \mathcal{M}_{*,a}^\infty(n)$.

*Proof.* This is a consequence of the definition of the space $\mathcal{M}_*^\infty(n)$, Corollary 9.13 and Lemma 9.5-(i). $\square$

According to Lemma 9.24, by taking the closure of the canonical image in $\mathcal{M}_*^\infty(n)$ as above, any strongly aperiodic complete connected Riemannian manifold of bounded geometry of infinite order can be realized as dense leaf of a compact foliated space with trivial holonomy groups. Since strong aperiodicity can be easily obtained with small perturbations of the metric tensor when the dimension is $\ge 2$, it follows that a complete connected Riemannian manifold is quasi-isometric with order $\infty$ to a leaf of a compact Riemannian foliated space if and only if it is of bounded geometry of order $\infty$. With more generality, we similarly get that a family $\mathcal{C}$ of complete connected Riemannian manifolds is equi-quasi-isometric with order $\infty$ to a collection of leaves of some compact Riemannian foliated space if and only if $\mathcal{C}$ is of equi-bounded geometry of order $\infty$. These properties are known and will be improved in Section ??.

To prove Theorem 9.15, recall first the following terminology and properties (see e.g. [?]). Let $M$ be a Riemannian manifold. A function $f: M \to \mathbb{R}$ is called *strictly convex* if its restriction to every nontrivial geodesic segment of $M$ is strictly convex—geodesic segments are identified with intervals in $\mathbb{R}$ via geodesic parametrizations. If $M$ is compact and $f$ is strictly convex, then $f$ reaches its minimum at a unique point in $M$. For a function $f \in C^2(M)$, the *Hessian* of $f$, denoted by Hess $f$, is the symmetric bilinear form on each tangent space of $M$ defined by

$$ \mathrm{Hess} f(\xi, \zeta) = g(\nabla_{\xi} \mathrm{grad} f, \zeta). $$

When $f$ is $C^2$, if Hess $f$ is positive definite, then $f$ is strictly convex. A domain $\Omega \subset M$ is said to be *strongly convex* when any pair of points $x, y \in \Omega$ is joined by a unique minimizing geodesic segment of $M$ which is contained in $\Omega$. Fix a strongly convex compact domain $\Omega$ in $M$. Let $\mathcal{C}(\Omega)$ be the set of functions $f \in C^2(\Omega)$ such that $\mathrm{grad} f$ is an outward pointing vector field on $\partial\Omega$ and Hess $f$ is positive definite on the interior $\mathrm{Int}(\Omega)$ of $\Omega$. Notice that $\mathcal{C}(\Omega)$ is open in the banach space $C^2(\Omega)$, and thus it is a $C^\infty$ banach manifold; moreover $\mathcal{C}(\Omega)$ is
preserved by the operations of sum and product by positive numbers. Any $f \in \mathcal{C}(\Omega)$ reaches its minimum at a unique point $m(f) \in \text{Int}(\Omega)$, defining a function $m : \mathcal{C}(\Omega) \to \text{Int}(\Omega)$.

**Lemma 9.25.** The function $m : \mathcal{C}(\Omega) \to \text{Int}(\Omega)$ is continuous.

*Proof.* Consider the map $v : \mathcal{C}(\Omega) \times \text{Int}(\Omega) \to T\Omega$ defined by $v(f, x) = \text{grad } f(x)$, and let $Z$ denote the zero section of $T\Omega$. Since the graph of the function $m$ is equal to $v^{-1}(Z)$, it is enough to prove that $v$ is $C^1$ and transverse to $Z$ (as a map between $C^\infty$ banach manifolds) [?, p. 45].

Let $TT\Omega = \mathcal{H} \oplus \mathcal{V}$ be the splitting into horizontal and vertical vector subbundles, and let $\pi_{\mathcal{H}}$ and $\pi_{\mathcal{V}}$ denote the corresponding projections of $TT\Omega$ onto $\mathcal{H}$ and $\mathcal{V}$. Let $\mathcal{X}^1(\Omega)$ denote the vector space of $C^1$ vector fields over $\Omega$, which is a banach space with the norm defined by

$$ \|X\| = \sup_{x \in \Omega} (|X(x)| + |\nabla X(x)|). $$

Notice that the gradient map $\text{grad} : C^2(\Omega) \to \mathcal{X}^1(\Omega)$ is a continuous linear map between banach spaces, and thus it is a $C^\infty$ map. Moreover the evaluation map $\text{ev} : \mathcal{X}^1(\Omega) \times \Omega \to T\Omega$, $\text{ev}(X, x) = X(x)$, is $C^1$; indeed, if $X \in \mathcal{X}^1(\Omega)$, $Y \in T_X \mathcal{X}^1(\Omega) = \mathcal{X}^1(\Omega)$, $x \in \Omega$ and $\xi \in T_x\Omega$, then $\text{ev}_*(Y, \xi) \in T_\xi T\Omega$ is easily seen to be determined by the conditions $\pi_{\mathcal{H}}(\text{ev}_*(Y, \xi)) = \xi$ in $\mathcal{H}_\xi = T_x\Omega$ and $\pi_{\mathcal{V}}(\text{ev}_*(Y, \xi)) = (Y + \nabla_\xi X)(x)$ in $\mathcal{V}_\xi = T_x\Omega$. Hence $v$ is a $C^1$ map because it is equal to the composite

$$ \mathcal{C}(\Omega) \times \text{Int}(\Omega) \hookrightarrow C^2(\Omega) \times \Omega \xrightarrow{\text{grad} \times \text{id}_{\Omega}} \mathcal{X}^1(\Omega) \times \Omega \xrightarrow{\text{ev}} T\Omega. $$

Fix any $f \in \mathcal{C}(\Omega)$ and $x \in \text{Int}(\Omega)$ with $v(f, x) \in Z$. Let $0_f$ and $0_x$ denote the zero tangent vectors of $\mathcal{C}(\Omega)$ and $\Omega$ at $f$ and $x$, respectively; thus $\text{grad } f(x) = 0_x$. According to the definition of covariant derivative, for any $\xi \in T_x\Omega$, we have

$$ \pi_{\mathcal{V}} \circ v_*(0_f, \xi) = \pi_{\mathcal{V}} \circ (\text{grad } f)_*(\xi) = \nabla_{\xi} \text{grad } f $$

via the canonical identity $\mathcal{V}_{0_x} = T_{x_x}\Omega$. But the mapping $\xi \mapsto \nabla_{\xi} \text{grad } f$ is an automorphism of $T_{x_x}\Omega$ because Hess $f$ is positive definite at $x$. It follows that $\pi_{\mathcal{V}} : v_*(\{0_f\} \times T_{x_x}\Omega) \to \mathcal{V}_{0_x}$ is an isomorphism, and thus $v_*(\{0_f\} \times T_{x_x}\Omega)$ is a linear complement of $\mathcal{H}_{0_x} = T_{0_x}Z$ in $T_{0_x}T\Omega$; moreover $v_*(\{0_f\} \times T_{x_x}\Omega)$ is closed in $T_{0_x}T\Omega$ because $T_{0_x}T\Omega$ is Hausdorff of finite dimension.

On the other hand, since $v_* : T_f\mathcal{C}(\Omega) \times T_{x_x}\Omega \to T_{0_x}T\Omega$ is linear and continuous, and $T_{0_x}T\Omega$ is Hausdorff of finite dimension, it follows that $(v_{*(f,x)}^{-1}(T_{0_x}Z))^{-1}(T_{0_x}Z)$ is a closed subspace of finite codimension of the banach space $T_f\mathcal{C}(\Omega) \times T_{x_x}\Omega$. Thus $(v_{*(f,x)}^{-1}(T_{0_x}Z))^{-1}(T_{0_x}Z)$ has a closed linear complement in $T_f\mathcal{C}(\Omega) \times T_{x_x}\Omega$ (see e.g. [?, p. 22]), which completes the proof of the transversality of $v$ to $Z$. $\square$

**Remarks.** (i) In the last part of the above proof, the space $(v_{*(f,x)}^{-1}(T_{0_x}Z))^{-1}(T_{0_x}Z)$ can be described as follows. Because $h \mapsto \text{grad } h(x)$ defines a continuous linear map $C^2(\Omega) \to T_{x_x}\Omega$, we get $v_*(T_f\mathcal{C}(\Omega) \times \{0_x\}) \subset \mathcal{V}_{0_x}$ and $v_*(h, 0_x) = \text{grad } h(x)$ by the canonical identity $\mathcal{V}_{0_x} = T_{x_x}\Omega$ for any $h \in C^2(\Omega) = T_f\mathcal{C}(\Omega)$, yielding

$$ (v_{*(f,x)})^{-1}(T_{0_x}Z) = \{(h, \xi) \in C^2(\Omega) \times T_{x_x}\Omega \mid \text{grad } h(x) + \nabla_{\xi} \text{grad } f = 0\}, $$
which is obviously closed and of finite codimension in $C^2(\Omega) \times T_x\Omega$.

(ii) Observe that, in the statement of Lemma 9.25, the map $m$ is $C^k$ if the banach space $C^{k+2}(\Omega)$ is used instead of $C^2(\Omega)$.

The main tool used to prove Theorem 9.15 is the Riemannian center of mass [?, ?], whose definition and main properties are recalled here. Suppose that the Riemannian manifold $M$ is connected and complete. Let $(A, \mu)$ be a probability space, and let $f: A \to M$ be a measurable map, which is called a mass distribution on $M$. Consider the function $P_f: M \to \mathbb{R}$ defined by

$$P_f(x) = \frac{1}{2} \int_A d(x, f(a))^2 \mu(a).$$

**Proposition 9.26** (H. Karcher). Assume that $f(\text{supp } \mu)$ is contained in some open ball $B$ of radius $r > 0$ in $M$ whose closure $\bar{B}$ is strongly convex. Then the following properties hold:

(i) The function $P_f$ is $C^\infty$ on $\bar{B}$, and $\text{grad } P_f$ is an outward pointing vector field on the boundary $\partial\bar{B}$.

(ii) If the sectional curvatures of $M$ in $B$ are $\le 0$, then Hess $P_f$ is positive definite on $B$.

(iii) If $\delta > 0$ is an upper bound for the sectional curvatures of $M$ in $B$, and $2r < \pi/2\sqrt{\delta}$, then Hess $P_f$ is positive definite on $B$.

According to Proposition 9.26, when the hypotheses of this result are satisfied, the function $P_f$ reaches its minimum on $\bar{B}$ at a unique point $C_f \in B$ (see e.g. [?]), which is called the center of mass of $f$.

It is proved in [?] that $C_f$ depends continuously on $f$ when $(A, \mu)$ is fixed, but indeed the following result follows directly from Lemma 9.25.

**Corollary 9.27.** The center of mass $C_f$ depends continuously on $f$, $\mu$ and the metric tensor of $M$.

**Lemma 9.28.** For any $n \in \mathbb{Z}_+$, any $[M, x] \in \mathcal{M}_{*,\text{la}}^\infty(n)$ and any $r > 0$ satisfying

$$\{h \in \text{Iso}(M) \mid h(x) \in \bar{B}(x, r)\} = \{\text{id}_M\},$$

there is some neighbourhood $\mathcal{L}$ of $[M, x]$ in $\mathcal{M}_{*,\text{la}}^\infty(n)$ such that

$$\{h \in \text{Iso}(L) \mid h(y) \in \bar{B}(y, r)\} = \{\text{id}_L\}$$

for all $[L, y] \in \mathcal{L}$.

*Proof.* Suppose that the statement is false. Then there is some sequence $[M_i, x_i]$ convergent to $[M, x]$ in $\mathcal{M}_*^\infty(n)$ so that, for each $i$, some $h_i \in \text{Iso}(M_i) \setminus \{\text{id}_{M_i}\}$ satisfies $h_i(x_i) \in \bar{B}(x_i, r)$. Choose any sequence of compact domains $\Omega_q$ of $M$ such that $\bar{B}(x, 2r) \subset \text{Int}(\Omega_q)$ and $d(x, \partial\Omega_q) \to \infty$ as $q \to \infty$. For each $q$ and $i$ large enough, there is some $C^\infty$ embedding $\phi_{q,i}: \Omega_q \to M_i$ so that $\phi_{q,i}(x) = x_i$ and $\phi_{q,i}^*g_i \to g|_{\Omega_q}$ as $i \to \infty$ with respect to the $C^\infty$ topology. Thus $\bar{B}(x_i, 2r) \subset \phi_{q,i}(\text{Int}(\Omega_q))$ for $i$ large enough.
**Claim 16.** If $r$ is small enough, we can assume that there is some $\delta > 0$ such that, for $i$ large enough, the maps $h_i$ can be chosen so that $d_i(z_i, h_i(z_i)) \ge \delta$ for some $z_i \in B(x_i, r)$.

Given any index $i$, suppose first that there is some $k \in \mathbb{Z} \setminus \{0\}$ such that $h_i^k(x_i) \notin \bar{B}(x_i, r/2)$. Then there is some $k \in \mathbb{Z} \setminus \{0\}$ such that $h_i^k(x_i) \notin \bar{B}(x_i, r/2)$ and $h_i^\ell(x_i) \in \bar{B}(x_i, r/2)$ if $|\ell| < |k|$. If $k = 1$, then $d_i(x_i, h_i(x_i)) \ge r/2$. If $k = -1$, then

$$d_i(x_i, h_i(x_i)) = d_i(h_i^{-1}(x_i), x_i) \ge r/2$$

as well. If $|k| \ge 2$, then there is some $\ell \in \mathbb{Z}$ such that $|\ell|, |k - \ell| < |k|$. Hence

$$ \begin{aligned} d_i(x_i, h_i^k(x_i)) &\le d_i(x_i, h_i^\ell(x_i)) + d_i(h_i^\ell(x_i), h_i^k(x_i)) \\ &= d_i(x_i, h_i^\ell(x_i)) + d_i(x_i, h_i^{k-\ell}(x_i)) \\ &\le r. \end{aligned} $$

Therefore, by using $h_i^k$ instead of $h_i$, we can assume that $d_i(x_i, h_i(x_i)) \ge r/2$ in this case.

Suppose otherwise that $h_i^k(x_i) \in \bar{B}(x_i, r/2)$ for all $k \in \mathbb{Z}$. Consider the non-trivial subgroup $A_i = \{h_i^k | k \in \mathbb{Z}\}$ of $\text{Iso}(M_i)$. Since $a(x_i) \in \bar{B}(x_i, r/2)$ for any $a \in A_i$, it follows that $A_i$ is compact by Corollary 8.3, and thus $A_i$ is a non-trivial compact lie subgroup of $\text{Iso}(M_i)$. Let $\mu_i$ be a bi-invariant probability measure on $A_i$. Let $f_i: A_i \to M$ be the mass distribution defined by $f_i(a) = a(x_i)$. By the $C^\infty$ convergence $\phi_{q,i}^*g_i \to g|_{\Omega_q}$, we can suppose that $r$ is so small that the ball $B(x_i, 2r/3)$ of $M_i$ satisfies the conditions of Proposition 9.26 for $i$ large enough. Then, since $f_i(A_i) \subset \bar{B}(x_i, r/2) \subset B(x_i, 2r/3)$, the center of mass $y_i = \mathbb{C}f_i$ is defined in $B(x_i, 2r/3)$. Moreover $y_i$ is easily seen to be a fixed point of the canonical action of $A_i$ on $M$ [?, Section 2]. Since there is a neighborhood of the identity in the orthogonal group $\text{O}(n)$ which contains no non-trivial subgroup (simply because $\text{O}(n)$ is a Lie group), it follows that there is some $K > 0$ such that, for any non-trivial subgroup $A \subset \text{O}(n)$, there is some $a \in A$ and some $v \in \mathbb{R}^n$ such that $|v| = 1$ and $|a(v) - v| \ge K$. In our setting, the subgroup $\{a_{*y_i} | a \in A_i\}$ of the orthogonal group $\text{O}(T_{y_i}M_i) \equiv \text{O}(n)$ is non-trivial because $M_i$ is connected and $A_i$ is non-trivial. Hence there is some $a_i \in A_i$ and some $\xi_i \in T_{y_i}M_i$ such that $|\xi_i| = 1$ and $|a_{i*}(\xi_i) - \xi_i| \ge K$. By the $C^\infty$ convergence $\phi_{q,i}^*g_i \to g|_{\Omega_q}$, we can also assume that $r$ is so small that there exists some equi-bi-Lipschitz distortion $C \ge 1$ of the exponential maps $\exp_{y_i}: B(0_{y_i}, r) \to B(y_i, r)$ for $i$ large enough. Then, for $z_i = \exp_{y_i}(\frac{1}{3}\xi_i) \in \bar{B}(y_i, r/3) \subset B(x_i, r)$, we get

$$d_i(z_i, a_i(z_i)) \ge \frac{r}{3C} |\xi_i - h_{i*}'(\xi_i)| \ge \frac{rK}{3C}.$$

Thus, by using $a_i$ instead of $h_i$, we can assume in this case that $d_i(z_i, h_i(z_i)) \ge rK/3C$. Therefore Claim 16 follows with $\delta = \min\{r/2, rK/3C\}$.

For each $q$, we can assume that

$$\bar{B}(x, \text{diam}(\Omega_q) + r) \subset \text{Int}(\Omega_{q+1}),$$