Proof. For each $\sigma \in \tilde{\mathcal{D}}_1$, the mapping $(\sigma_1, \dots, \sigma_n) \mapsto (\bar{\sigma}_n, \dots, \bar{\sigma}_1)$ defines a bijection between the set of elements $(\sigma_1, \dots, \sigma_n) \in \tilde{\mathcal{D}}_\infty$ satisfying $\sigma_n \cdots \sigma_1 = \sigma$ and the set of elements $(\tau_1, \dots, \tau_n) \in \tilde{\mathcal{D}}_\infty$ satisfying $\tau_n \cdots \tau_1 = \bar{\sigma}$. Hence the result follows by (10.31). $\square$

$\mathcal{C}_0$ and $\mathcal{C} \setminus \mathcal{C}_0$ will be endowed with the pseudo-metrics with possible infinite values, denoted by $d_b$, given by (10.16) and Proposition 10.36. But $b$ also induces a pseudo-metric with possible infinite values on $\bar{\mathcal{C}}_0$, whose construction needs the following. Let $\mathcal{J}$ be a small weakly filtered category, let $\mathcal{F} = \tilde{\mathcal{D}}_{\text{stp}}^J$ with the involution induced by the involution of $\mathcal{D}$, and let $\mathcal{E}$ be the involutive subcategory of $\mathcal{F}$ given by the a-natural transformations between Cauchy stp-maps $\mathcal{J} \to \tilde{\mathcal{D}}$. Let $\tilde{\mathcal{F}}$ be the lift of $\mathcal{F}$ by the inclusion u-functor $\mathcal{E}/\mathcal{F}$, and let $b^J: \tilde{\mathcal{F}} \to [0, \infty]$ be defined by

$$b^J(t) = \limsup_{i,j \to \infty} \bigcup_{f \in \mathcal{J}(i,j)} \{F'(f) t_i, t_j F(f)\}^\times.$$

By (c2),

$$(10.33) \qquad b^J(t) \ge \limsup_{i \to \infty} b(t_i).$$

**Proposition 10.85.** $b^J$ is a distortion of $\mathcal{E}/\mathcal{F}$.

Proof. It is obvious that $b^J$ satisfies (d1). To check (d2), take a finite sequence of consecutive morphisms of $\tilde{\mathcal{F}}$,

$$F^0 \xrightarrow{t^1} F^1 \xrightarrow{t^2} \dots \xrightarrow{t^n} F^n.$$

For $i, j \in \mathcal{J}_0$ and $f \in \mathcal{J}(i, j)$, by applying Lemma 10.68 to the diagram

$$\begin{tikzcd}
F_i^0 \arrow[r, "t_i^1"] \arrow[d, "F^0(f)"'] & F_i^1 \arrow[d, "F^1(f)"'] & \dots \\
F_j^0 \arrow[r, "t_j^1"] \arrow[d, "F^0(f)"'] & F_j^1 \arrow[d, "F^1(f)"'] &
\end{tikzcd}
\arrow[r, "F_i^{n-1}", "t_i^n"] \arrow[r, "F_j^{n-1}", "t_j^n"]
\arrow[d, "F^{n-1}(f)", "\dots"] \arrow[d, "F^n(f)", "\dots"]
$$

we get

$$\{F^n(f) t_i^n \dots t_i^1, t_j^n \dots t_j^1 F^0(f)\}^\times \asymp \{F^1(f) t_i^1, t_j^1 F^0(f)\}^\times \cup \dots \cup \{F^n(f) t_i^n, t_j^n F^{n-1}(f)\}^\times \\ \cup \{t_i^1 + \dots + t_i^n, t_j^1 + \dots + t_j^n\},$$

yielding

$$b^J(t^n \cdots t^1) \asymp b^J(t^n) + \cdots + b^J(t^1)$$

by (10.33). $\square$

Let $\mathcal{J}$ be another small weakly filtered category. The following result is elementary.

**Lemma 10.86.** For any $t \in \tilde{\mathcal{D}}_{\text{stp}}^J$ and $u \in \tilde{\mathcal{D}}_{\text{stp}}^J$, if $u$ is a cofinal sub-transformation of $t$, then $b^J(u) \le b^J(t)$.
Let $\hat{\mathfrak{b}}^J$ denote the distortion of $\mathcal{E}/\mathcal{F}$ induced by $\mathfrak{b}^J$ according to (10.14). For $F, F' \in \bar{\mathbb{C}}_0$ with possibly different source categories, let $\Lambda_0(F, F')$ be the set of pairs $(A, A')$, where $A$ and $A'$ are cofinal sub-a-functors of $F$ and $F'$, respectively, with the same source category. Moreover let $\Lambda_1(F, F')$ be the family of transformations $A \to A'$ with $(A, A')$ running in $\Lambda_0(F, F')$, and let $\Lambda_\infty(F, F')$ be the family of finite sequences of consecutive small weakly filtered transformations between objects of $\bar{\mathbb{C}}_0$ with the same source category,

$$A^0 \xrightarrow{t^1} A^1 \xrightarrow{t^2} \dots \xrightarrow{t^m} A^m,$$

such that $(A^0, A^m) \in \Lambda_0(F, F')$. Observe that $\Lambda_1(F, F') \neq \emptyset$ by Lemma 10.79. With the same norm on $\mathbb{R}^2$ as above, let

$$\bar{d}_b(F, F') = \inf\{\|(\inf \hat{\mathfrak{b}}_{A,A'}^J, \inf \hat{\mathfrak{b}}_{A',A}^J)\| \mid (A, A') \in \Lambda_0(F, F')\}$$

in $[0, \infty]$, where $J$ denotes the source category of $A$ and $A'$. Notice that

$$(10.34) \qquad \bar{\mathbb{C}}(F, F') \neq \emptyset \implies \bar{d}_b(F, F') = 0.$$

**Proposition 10.87.** $\bar{d}_b$ is a pseudo-metric with possible infinite values on $\bar{\mathbb{C}}_0$ invariant by the canonical action of $\bar{\mathbb{C}}$.

*Proof.* The symmetry of $\bar{d}_b$ is obvious. Moreover we get $\bar{d}_b(F, F) = 0$ for any $F \in \bar{\mathbb{C}}_0$ by considering the identity transformation of $F$.

To prove the triangle inequality, take $F, F', F'' \in \bar{\mathbb{C}}_0$ with source categories $J, J'$ and $J'',$ respectively. Consider finite sequence of consecutive small weakly filtered transformations,

$$A^0 \xrightarrow{t^1} A^1 \xrightarrow{t^2} \dots \xrightarrow{t^m} A^m, \quad B^0 \xrightarrow{u^1} B^1 \xrightarrow{u^2} \dots \xrightarrow{u^n} B^n,$$

defining elements of $\Lambda_\infty(F, F')$ and $\Lambda_\infty(F', F'')$, respectively. The source category of $A^0, \dots, A^m$ is denoted by $J$, and the source category of $B^0, \dots, B^m$ is denoted by $K$. Let $\mathbf{a}^m : J \to J'$ and $\mathbf{b}^0 : K \to J'$ be cofinal stp-reparametrizations defining $A^m$ and $B^0$, respectively. By Lemma 10.79, for some weakly filtered category $\mathcal{X}$, there are cofinal functors $\mathbf{p} : \mathcal{X} \to J$ and $\mathbf{q} : \mathcal{X} \to K$, and there is a transformation $\xi : \mathbf{a}^m \mathbf{p} \to \mathbf{b}^0 \mathbf{q}$. For $x, y \in \mathcal{X}_0$ and $f \in \mathcal{X}(x, y)$,

$$\{B^0 \mathbf{q}(f) (F'\xi)_x, (F'\xi)_y A^m \mathbf{p}(f)\}^\times = \{F' \mathbf{b}^0 \mathbf{q}(f) F'(\xi_x), F'(\xi_y) F' \mathbf{a}^m \mathbf{p}(f)\}^\times,$$

which converges to zero as $x, y \to \infty$ by Lemma 10.70, and since $F'$ is a-commutative a-functor, and $\mathbf{a}^m \mathbf{p}$ and $\mathbf{b}^0 \mathbf{q}$ are cofinal. This shows that $\mathbf{b}^\chi(F'\xi) = 0$. So the sequence of consecutive small weakly filtered transformations

$$A^0 \mathbf{p} \xrightarrow{t^1 \mathbf{p}} \dots \xrightarrow{t^m \mathbf{p}} A^m \mathbf{p} \xrightarrow{F' \xi} B^0 \mathbf{q} \xrightarrow{u^1 \mathbf{q}} \dots \xrightarrow{u^n \mathbf{q}} B^n \mathbf{q}$$

defines an element of $\Lambda_\infty(F, F'')$ and satisfies

$$\begin{aligned} & \mathbf{b}^\chi(t^1 \mathbf{p}) + \dots + \mathbf{b}^\chi(t^m \mathbf{p}) + \mathbf{b}^\chi(F'\xi) + \mathbf{b}^\chi(u^1 \mathbf{q}) + \dots + \mathbf{b}^\chi(u^n \mathbf{q}) \\ & \leq \mathbf{b}^J(t^1) + \dots + \mathbf{b}^J(t^m) + \mathbf{b}^K(u^1) + \dots + \mathbf{b}^K(u^n) \end{aligned}$$
by Lemma 10.86. Hence

$$ \inf \hat{\mathfrak{b}}_{A^0, B^n q}^{\chi} \leq \inf \hat{\mathfrak{b}}_{A^0, A^m}^{\mathcal{J}} + \inf \hat{\mathfrak{b}}_{B^0, B^n}^{\mathcal{K}}. $$

Similarly,

$$ \inf \hat{\mathfrak{b}}_{B^n q, A^0 p}^{\chi} \leq \inf \hat{\mathfrak{b}}_{B^n, B^0}^{\mathcal{K}} + \inf \hat{\mathfrak{b}}_{A^m, A^0}^{\mathcal{J}}, $$

yielding

$$ \bar{d}_{\mathfrak{b}}(F, F'') \leq \bar{d}_{\mathfrak{b}}(F, F') + \bar{d}_{\mathfrak{b}}(F', F'') $$

by (10.15).

The $\overline{\mathbb{C}}$-invariance of $\bar{d}_{\mathfrak{b}}$ follows from its triangle inequality and (10.34). $\square$

By Proposition 10.87, $\bar{d}_{\mathfrak{b}}$ induces a pseudo-metric with possible infinite values on $\overline{\mathbb{C}}\backslash\overline{\mathbb{C}}_0$, also denoted by $\bar{d}_{\mathfrak{b}}$. From now on, $\overline{\mathbb{C}}_0$ and $\overline{\mathbb{C}}\backslash\overline{\mathbb{C}}_0$ will be endowed with these pseudo-metrics with possible infinite values.

**Proposition 10.88.** $\iota: \mathcal{C}_0 \to \overline{\mathbb{C}}_0$ is isometric.

*Proof.* Let $M, N \in \mathcal{C}_0$. For any sequence of consecutive morphisms of $\tilde{\mathcal{D}}$ of the form

$$ M = L_0 \xrightarrow{f^1} L_1 \xrightarrow{f^2} \dots \xrightarrow{f^m} L_m = M, $$

we get a sequence of transformations

$$ \iota_M = \iota_{L_0} \xrightarrow{t^1} \iota_{L_1} \xrightarrow{t^2} \dots \xrightarrow{t^m} \iota_{L_m} = \iota_M, $$

where $t_O^k = f^k$ for $k \in \{1, \dots, m\}$. These transformations are natural, but they may not be a-natural because they may not be undistorted; indeed, $\mathfrak{b}^0(t^k) = \mathfrak{b}(f^k)$. Then

$$ \inf \hat{\mathfrak{b}}_{\iota_M, \iota_N}^{\mathcal{O}} \leq \mathfrak{b}^{\mathcal{O}}(t^k) + \dots + \mathfrak{b}^{\mathcal{O}}(t^m) = \mathfrak{b}(f^1) + \dots + \mathfrak{b}(f^m), $$

yielding $\inf \hat{\mathfrak{b}}_{\iota_M, \iota_N}^{\mathcal{O}} \leq \inf \hat{\mathfrak{b}}_{M,N}$. Similarly, $\inf \hat{\mathfrak{b}}_{\iota_N, \iota_M}^{\mathcal{O}} \leq \inf \hat{\mathfrak{b}}_{N,M}$. So $\bar{d}_{\mathfrak{b}}(\iota_M, \iota_N) \leq d_{\mathfrak{b}}(M, N)$ by (10.15).

Now, let

$$ A^0 \xrightarrow{t^1} A^1 \xrightarrow{t^2} \dots \xrightarrow{t^m} A^m $$

be a finite sequence of consecutive small weakly filtered transformations defining an element of $\Lambda_{\infty}(\iota_M, \iota_N)$, and let $\mathcal{J}$ denote the source category of $A^0, \dots, A^m$. Since $A_j^0 = M$ and $A_j^m = N$ for all $j \in \mathcal{J}_0$, we get

$$ \inf \hat{\mathfrak{b}}_{M,N} \leq \mathfrak{b}(t_j^1) + \dots + \mathfrak{b}(t_j^m) $$

by considering the consecutive morphisms

$$ M = A_j^0 \xrightarrow{t_j^1} A_j^1 \xrightarrow{t_j^2} \dots \xrightarrow{t_j^m} A_j^m = N $$
in $\tilde{\mathcal{B}}$. So

$$
\begin{align*}
\inf \hat{\mathfrak{b}}_{M,N} &\le \limsup_{j \to \infty} (\mathfrak{b}(t_j^1) + \dots + \mathfrak{b}(t_j^m)) \\
&\le \limsup_{j \to \infty} \mathfrak{b}(t_j^1) + \dots + \limsup_{j \to \infty} \mathfrak{b}(t_j^m) \\
&\le \mathfrak{b}^j(t^1) + \dots + \mathfrak{b}^j(t^m)
\end{align*}
$$

by (10.33), yielding inf $\hat{\mathfrak{b}}_{M,N} \le \inf \hat{\mathfrak{b}}_{\iota_M, \iota_N}^j$. Similarly, inf $\hat{\mathfrak{b}}_{N,M} \le \inf \hat{\mathfrak{b}}_{\iota_N, \iota_M}^j$. Thus $d_{\mathfrak{b}}(M,N) \le \bar{d}_{\mathfrak{b}}(\iota_M, \iota_N)$ by (10.15). $\square$

**Proposition 10.89.** $\overline{\mathbb{C}}_0$ is complete.

*Proof.* Let ($F^n$) be a Cauchy sequence in $\overline{\mathbb{C}}_0$, and let $\mathcal{J}^n$ be the source category of each $F^n$. By choosing a subsequence if necessary, we can assume that

$$
\bar{d}_{\mathfrak{b}}(F^n, F^{n+1}) < \frac{1}{2^n}
$$

for each $n \in \mathbb{N}$. Then, by (d2) for $\mathfrak{b}^j$, there is some transformation $t^n : A^n \to B^n$ in
$\Lambda_1(F^n, F^{n+1})$ such that

$$
(10.35) \qquad \mathfrak{b}^{\mathcal{J}^n}(t^n) < \frac{1}{2^n},
$$

where $\mathcal{J}^n$ is the same source category of $A^n$ and $B^n$. Let $\mathbf{a}^n : \mathcal{J}^n \to \mathcal{J}^n$ and $\mathbf{b}^n : \mathcal{J}^n \to \mathcal{J}^{n+1}$ be cofinal stp-reparametrizations defining $A^n$ and $B^n$, respectively. By Lemma 10.70 and since $F^n$ is an a-commutative a-functor, there is some $i_n^* \in \mathcal{J}_0^n$ such that

$$
(10.36) \quad \bigcup_{i_n^* \le i, i' \atop i_n^* \le i''} \left( \bigcup_{i_n^* \le i, i' \atop i_n^* \le i''} \left( F^n(i'', i') F^n(i, i'') \cup \overline{F^n(i', i'')} F^n(i, i'') \cup F^n(i'', i') \overline{F^n(i'', i)} \right) \right)^\times < \frac{1}{2^n}.
$$

By (10.35), there is some $j_n^* \in \mathcal{J}_0^n$ for each $n \in \mathbb{N}$ such that $i_n^* \le \mathbf{a}_{j_n^*}^n, i_{n+1}^* \le \mathbf{b}_{j_n^*}^n$ and

$$
(10.37) \qquad \bigcup_{j_n^* \le j, j' \atop p \in \mathcal{J}^n(j, j')} \{A^n(p) t_j^n, t_{j'}^n B^n(p)\}^\times < \frac{1}{2^n}.
$$

Let $\hat{\mathcal{J}}^n$ and $\hat{\mathcal{G}}^n$ be the cofinal full subcategories of $\mathcal{J}^n$ and $\mathcal{G}^n$, respectively, with

$$
\hat{\mathcal{J}}_0^n = \{i \in \mathcal{J}_0^n \mid i_n^* \le i\}, \quad \hat{\mathcal{G}}_0^n = \{j \in \mathcal{G}_0^n \mid j_n^* \le j\}.
$$

Let $\mathcal{K}$ be the weakly filtered small category defined in the following way:

$$
\bullet \mathcal{K}_0 = \bigsqcup_{n=1}^{\infty} \hat{\mathcal{J}}_0^n = \bigsqcup_{n=1}^{\infty} (\{n\} \times \hat{\mathcal{J}}_0^n).
$$

* For $(n, i), (n', i') \in \mathcal{K}_0$ with $n > n'$, let $\mathcal{K}(n, i; n', i') = \emptyset$.
• For $(n, i), (n', i') \in \mathcal{K}_0$ with $n \le n'$, $\mathcal{K}(n, i; n', i')$ is the set of sequences of the form

$$ (f_{n'}, j_{n'-1}, f_{n'-1}, \dots, f_{n+1}, j_n, f_n), $$

where, for $m \in \{n, n+1, \dots, n'\}$, $f_m \in \mathcal{I}^k(i_m, i'_m)$ for some $i_m, i'_m \in \hat{\mathcal{I}}_0^m$, and $j_m \in \hat{\mathcal{J}}_0^m$ if $m < n'$, and such that $i_n = i, i'_{n'} = i', \text{ and } \mathbf{a}_{j_m}^m = i'_m$ and $\mathbf{b}_{j_m}^m = i_{m+1}$ if $m < n'$.

• For $(n, i), (n', i'), (n'', i'') \in \mathcal{K}_0$ with $n \le n' \le n''$ and

$$ \begin{aligned} \lambda &= (f_{n'}, j_{n'-1}, f_{n'-1}, \dots, f_{n+1}, j_n, f_n) \in \mathcal{K}(n, i; n', i'), \\ \mu &= (g_{n''}, j_{n''-1}, g_{n''-1}, \dots, g_{n'+1}, j_{n'}, g_{n'}) \in \mathcal{K}(n', i'; n'', i''), \end{aligned} $$

let

$$ \mu\lambda = (g_{n''}, j_{n''-1}, g_{n''-1}, \dots, g_{n'+1}, j_{n'}, g_{n'}f_{n'}, j_{n'-1}, f_{n'-1}, \dots, f_{n+1}, j_n, f_n). $$

Observe that each $\hat{\mathcal{I}}^n$ is canonically isomorphic to the full subcategory of $\mathcal{K}$ whose set of units is $\{n\} \times \hat{\mathcal{I}}_0^n$, which is not cofinal.

Let $G: \mathcal{K} \to \tilde{\mathcal{D}}$ be the stp-map defined by $G_{n,i} = F_i^n$ for $(n,i) \in \mathcal{K}_0$, and

$$ G(f_{n'}, j_{n'-1}, f_{n'-1}, \dots, f_{n+1}, j_n, f_n) = F^{n'}(f_{n'}) t_{j_{n'-1}}^{n'-1} F^{n'-1}(f_{n'-1}) \cdots F^{n+1}(f_{n+1}) t_{j_n}^n F^n(f_n) $$

for $(n, i), (n', i') \in \mathcal{K}_0$ and

$$ (f_{n'}, j_{n'-1}, f_{n'-1}, \dots, f_{n+1}, j_n, f_n) \in \mathcal{K}(n, i; n', i'). $$

**Claim 30.** *G is a-commutative.*

This assertion follows by showing that

$$ (10.38) \qquad \lim_{n,n' \to \infty} \bigcup_{i \in \mathcal{I}_0^n, i' \in \mathcal{I}_0^{n'}} G(\mathcal{K}(n, i; n', i'))^\times = 0. $$

We can assume $n \le n'$ to prove this convergence. Since $G$ can be identified to the restriction of $F^n$ on

$$ \mathcal{K}(n, i, n, i') = \{n\} \times \hat{\mathcal{I}}^n(i, i') \equiv \hat{\mathcal{I}}^n(i, i') $$

for $i \in \mathcal{I}_0^n$ and $i' \in \mathcal{I}_0^{n'}$, (10.38) is given by (10.36) when $n' = n$.

Now suppose that $n' > n$. Fix some $i \in \mathcal{I}_0^n$ and some $i' \in \mathcal{I}_0^{n'}$, and take two morphisms in $\mathcal{K}(n, i; n', i')$,

$$ \lambda = (f_{n'}, j_{n'-1}, f_{n'-1}, \dots, f_{n+1}, j_n, f_n), $$

$$ \mu = (g_{n'}, k_{n'-1}, g_{n'-1}, \dots, g_{n+1}, k_n, g_n). $$
Since $\mathbf{a}^m$ and $\mathbf{b}^m$ are cofinal, there is some $\ell_m \in \hat{\mathcal{J}}_0^m$ for each $m \in \{n, \dots, n' - 1\}$ such that $j_m, k_m \le \ell_m$, and $\mathbf{b}_{\ell_{m-1}}^{m-1} \le \mathbf{a}_{\ell_m}^m$ if $m > n$ (when $n' > n + 1$). Then there are morphisms $p_m \in \hat{\mathcal{J}}^m(j_m, \ell_m)$ and $q_m \in \hat{\mathcal{J}}^m(k_m, \ell_m)$ for all $m$, and $h_m \in \hat{\mathcal{J}}^m(\mathbf{b}_{\ell_{m-1}}^{m-1}, \mathbf{a}_{\ell_m}^m)$ if $m > n$. For the sake of simplicity, the superscript $m$ will be removed from $F^m$, $\mathbf{a}^m$, $\mathbf{b}^m$ and $t^m$ in the rest of the proof of (10.38) because there is no possibility of confusion. The above morphisms form the diagram

<figure><img src="image_31.png" /></figure>

Observe that the composites of the top and bottom rows of this diagram are $G(\lambda)$ and $G(\mu)$, respectively. By Lemma 10.68,

$$
\begin{align*}
\{G(\lambda), G(\mu)\}^\times &\asymp \{\overline{\mathbf{a}(q_n)} \mathbf{a}(p_n) F(f_n), F(g_n)\}^\times \\
&\cup \bigcup_{m=n}^{n'-1} \{\overline{\mathbf{b}(q_m)} \mathbf{b}(p_m) t_{j_m}, t_{k_m} \overline{\mathbf{a}(q_m)} \mathbf{a}(p_m)\}^\times \\
&\cup \bigcup_{m=n+1}^{n'-1} \{\overline{\mathbf{a}(q_m)} \mathbf{a}(p_m) F(f_m), F(g_m) \overline{\mathbf{b}(q_{m-1})} \mathbf{b}(p_{m-1})\}^\times \\
&\cup \{F(f_{n'}), F(g_{n'}) \overline{\mathbf{b}(q_{n'-1})} \mathbf{b}(p_{n'-1})\}^\times \\
&\cup \{F(f_n) + \dots + F(f_{n'}) + t_{j_n} + \dots + t_{j_{n'-1}}, \\
&\phantom{\cup \{F(f_n) + \dots + F(f_{n'}) + t_{j_n} + \dots + t_{j_{n'-1}},} F(g_n) + \dots + F(g_{n'}) + t_{k_n} + \dots + t_{k_{n'-1}}\}.
\end{align*}
$$
On the other hand, by (b1), (10.31) and Corollary 10.67,

$$
\begin{align*}
\{\overline{Fa}(q_n) Fa(p_n) F(f_n), F(g_n)\}^\times &\approx \{\overline{F(f_n) Fa(p_n) Fa(q_n), F(g_n)}\}^\times \\
&\asymp \{\overline{F(f_n) Fa(p_n), F(g_n) Fa(q_n)}\}^\times \cup \{Fa(q_n), F(g_n)\} \\
&\approx \{Fa(p_n) F(f_n), Fa(q_n) F(g_n)\}^\times \cup \{Fa(q_n), F(g_n)\}, \\
\{F(f_{n'}), F(g_{n'}) \overline{Fb(q_{n'-1}) Fb(p_{n'-1})}\}^\times &\asymp \{F(f_{n'}) \overline{Fb(p_{n'-1})}, F(g_{n'}) \overline{Fb(q_{n'-1})}\}^\times \\
&\phantom{\asymp} \cup \{Fb(p_{n'-1}), F(f_{n'})\}.
\end{align*}
$$

Moreover, by Lemmas 10.68 and (10.31),

$$
\begin{align*}
\{\overline{Fb}(q_m) Fb(p_m) t_{j_m}, t_{k_m} \overline{Fa}(q_m) Fa(p_m)\}^\times
  &\asymp \{Fb(p_m) t_{j_m}, t_{l_m} Fa(p_m)\}^\times \cup \{\overline{Fb}(q_m) t_{l_m}, t_{k_m} \overline{Fa}(q_m)\}^\times \\
  &\phantom{\asymp} \cup \{Fa(p_m), Fa(q_m), Fb(p_m), Fb(q_m)\}
\end{align*}
$$

for $m \in \{n, \dots, n' - 1\}$, and

$$
\begin{align*}
\{\overline{Fa}(q_m) Fa(p_m) F(f_m), F(g_m) \overline{Fb}(q_{m-1}) Fb(p_{m-1})\}^\times
  &\asymp \{Fa(p_m) F(f_m), F(h_m) Fb(p_{m-1})\}^\times \cup \{\overline{Fa}(q_m) F(h_m), F(g_m) \overline{Fb}(q_{m-1})\}^\times \\
  &\phantom{\asymp} \cup \{Fb(p_{m-1}), Fb(q_{m-1}), Fa(p_m), Fa(q_m)\}
\end{align*}
$$

for $m \in \{n+1, \dots, n' - 1\}$. Therefore, by (10.36) and (10.37),

$$
\{G(\lambda), G(\mu)\}^\times \asymp \frac{1}{2^n} + \dots + \frac{1}{2^{n'}},
$$

which converges to zero as $n, n' \to \infty$ uniformly on $\lambda$ and $\mu$, yielding (10.38) by (b5).

**Claim 31.** *G is an a-functor.*

To prove this assertion, take $\lambda$ and $\mu$ like in the definition of the operation of $\mathcal{K}$, and let
$i'_1 = \mathbf{b}_{j_{n'-1}}^{n'-1}, i'_2 = \mathbf{a}_{j_{n'}}^{n'}$, and

$$
\begin{align*}
\lambda_0 &= (\mathrm{id}_{i'_1}, j_{n'-1}, f_{n'-1}, \dots, f_{n+1}, j_n, f_n) \in \mathcal{K}(n, i; n', i'_1), \\
\mu_0 &= (g_{n''}, j_{n''-1}, g_{n''-1}, \dots, g_{n'+1}, j_{n'}, \mathrm{id}_{i'_2}) \in \mathcal{K}(n', i'_2; n'', i'') .
\end{align*}
$$

Notice that

$$
\begin{align*}
G(\lambda) &= F^{n'}(f_{n'}) G(\lambda_0), \quad G(\mu) = G(\mu_0) F^{n'}(g_{n'}), \\
G(\mu\lambda) &= G(\mu_0) F^{n'}(g_{n'} f_{n'}) G(\lambda_0).
\end{align*}
$$

Hence, by Lemma 10.64,

$$
\{G(\mu\lambda), G(\mu)G(\lambda)\}^\times \asymp \{F^{n'}(g_{n'}f_{n'}), F^{n'}(g_{n'})F^{n'}(f_{n'})\}^\times \cup \{G(\mu_0), G(\lambda_0)\},
$$
which converges to zero uniformly on $\lambda$ and $\mu$ as $n, n', n'' \to \infty$ by (10.36) and (10.38), completing the proof of Claim 31.

Now, for each $m \in \mathbb{N}$, consider the full subcategory $\mathcal{X}^m$ of $(\prod_{k=1}^\infty \hat{\mathcal{D}}^k) \times \mathcal{K}$ with $\mathcal{X}_0$ equal to the set of objects $((j_k); n, i) \in (\prod_{k=1}^\infty \hat{\mathcal{D}}_0^k) \times \mathcal{K}_0$ such that $2, m \le n$, $\mathbf{b}_{j_k}^k \le \mathbf{a}_{j_{k+1}}^{k+1}$ in $\hat{\mathcal{J}}_0^{k+1}$ for all $k \in \{1, \dots, n-2\}$ if $n \ge 3$, and $\mathbf{b}_{j_{n-1}}^{n-1} = i$ in $\hat{\mathcal{J}}_0^n$. Let $\mathbf{p}^m: \mathcal{X}^m \to \mathcal{J}^m$ and $\mathbf{q}^m: \mathcal{X}^m \to \mathcal{K}$ be the functors given by the factor projections, which are cofinal by the cofinality of the stp-maps $\mathbf{a}^k$ and $\mathbf{b}^k$. Moreover let $\mathbf{r}^m = \mathbf{a}^m\mathbf{p}^m: \mathcal{X}^m \to \mathcal{J}^m$. Then $R^m = F^m\mathbf{r}^m$ and $Q^m = G\mathbf{q}^m$ are small a-commutative sub-a-functors of $F^m$ and $G$, respectively, because they are cofinal sub-stp-maps, and $F^m$ and $G$ are commutative a-functors (Claims 30 and 31).

For each $k \in \mathbb{N}$, $j \in \hat{\mathcal{J}}_0^k$ and $j' \in \hat{\mathcal{J}}^{k+1}$ so that $\mathbf{b}_j^k \le \mathbf{a}_{j'}^{k+1}$ in $\hat{\mathcal{J}}_0^{k+1}$, fix some morphism $\xi_{j,j'}^{k+1} \in \hat{\mathcal{J}}^{k+1}(\mathbf{b}_j^k, \mathbf{a}_{j'}^{k+1})$. Then let $u^m: R^m \to Q^m$ be the transformation defined by

$$u_x^m = t_{j_{n-1}}^{n-1} F^{n-1}(\xi_{j_{n-1},j_{n-2}}^{n-1}) t_{j_{n-2}}^{n-2} \cdots t_{j_{m+1}}^{m+1} F^{m+1}(\xi_{j_{m+1},j_m}^{m+1}) t_{j_m}^m$$

for $x = ((j_k); n, i) \in \mathcal{X}_0^m$; observe that $u_x^m$ is a morphism of $\tilde{\mathcal{D}}$ with source $A_{j_m}^m = R_x^m$ and target $B_{j_{n-1}}^{n-1} = F_i^n = G_{n,i} = Q_x^m$.

**Claim 32.** $\mathbf{b}^{\mathcal{X}^m}(u^m) \to 0$ as $m \to \infty$.

For $x, y \in \mathcal{X}^m$ and $\theta \in \mathcal{X}^m(x, y)$, notice that

$$\{R^m(\theta) u_x, u_y Q^m(\theta)\} \subset G(\mathcal{K}(m, \mathbf{r}_x^m; \mathbf{q}_y^m)).$$

Thus Claim 32 also follows from (10.38).

Finally, $G \in \bar{\mathcal{C}}_0$ by Claims 30 and 31, and $F^m \to G$ as $m \to \infty$ by Claim 32 and Lemma 10.84. $\square$

**Proposition 10.90.** $\iota(\mathcal{C}_0)$ is dense in $\bar{\mathcal{C}}_0$.

*Proof.* Let $F \in \bar{\mathcal{C}}_0$ with source category $\mathcal{J}$. Let us show that $\iota_{F_i} \to F$ in $\bar{\mathcal{C}}_0$ as $i \to \infty$ in $\mathcal{J}_0$. For this purpose, given any $i \in \mathcal{J}_0$, let $\mathcal{J}^i$ be the cofinal full subcategory of $\mathcal{J}$ with $\mathcal{J}_0^i = \{j \in \mathcal{J}_0 | i \le j\}$, let $\mathbf{a}^i: \mathcal{J}^i \to \mathcal{O}$ be the trivial functor, and let $\mathbf{b}^i: \mathcal{J}^i \to \mathcal{J}$ be the inclusion functor. Then $A^i = \iota_{F_i}\mathbf{a}^i$ and $B^i = F\mathbf{b}^i$ are cofinal sub-a-functors of $\iota_{F_i}$ and $F$, respectively. For every $j \in \mathcal{J}_0^i$, fix any $\xi_{i,j} \in \mathcal{J}(i, j)$, and let $t^i: A^i \to B^i$ be the transformation defined by $t_j^i = F(\xi_{i,j}) \in \tilde{\mathcal{D}}(F_i, F_j) = \tilde{\mathcal{D}}(A_j^i, B_j^i)$. For $j, j' \in \mathcal{J}_0^i$ and $f \in \mathcal{J}^i(j, j') = \mathcal{J}(j, j')$,

$$\{B^i(f) t_j^i, t_{j'}^i A^i(f)\}^\times = \{F(f) F(\xi_{i,j}), F(\xi_{i,j'})\}^\times,$$

which converges to zero as $i \to \infty$ uniformly on $f, j$ and $j'$ by Lemma 10.70 since $F$ is an a-commutative a-functor. Hence $\mathbf{b}^{\mathcal{J}^i}(t^i) \to 0$ as $i \to \infty$, yielding $\bar{d}_b(\iota_{F_i}, F) \to 0$ as $i \to \infty$ by Lemma 10.84. $\square$

**Definition 10.91.** It is said that $\mathbf{b}$ is complete if there is some stp-map $\lim: \bar{\mathcal{C}} \to \mathcal{C}$, called limit, such that there are transformations $\lim \circ \iota \to \mathrm{id}_{\mathcal{C}}$ and $\iota \circ \lim \to \mathrm{id}_{\bar{\mathcal{C}}}$.
The condition of Definition 10.91 is weaker than requiring $\iota$ to be a category equivalence. If $\iota$ is a category equivalence, then $\mathcal{C}$ is a groupoid since $\bar{\mathcal{C}}$ is a groupoid.

**Proposition 10.92.** *b* is complete if and only if $\bar{\iota}$ is bijective.

*Proof.* The “only if” part is elementary. To prove the “if” part, suppose that $\bar{\iota}$ is a bijection. Then, for each $F \in \bar{\mathcal{C}}_0$, fix any $\lim_F \in \bar{\iota}^{-1}([F]) \in \mathcal{C}\bar{\mathcal{C}}_0$. Let $[t] : F \to F'$ be a morphism of $\bar{\mathcal{C}}$. Then $[F] = [F']$ in $\bar{\mathcal{C}}\bar{\mathcal{C}}_0$, and thus $\bar{\iota}^{-1}([F]) = \bar{\iota}^{-1}([F'])$. This means that $\mathcal{C}(\lim_F, \lim_{F'}) \neq \emptyset$, and we can define $\lim([t])$ to be any fixed morphism $\lim_F \to \lim_{F'}$ of $\mathcal{C}$. These choices define an stp-map $\bar{\mathcal{C}} \to \mathcal{C}$. For $M \in \mathcal{C}_0$, since $(\lim \circ \iota)_M \in \bar{\iota}^{-1}(\iota_M)$, we can choose some $\xi_M \in \mathcal{C}((\lim \circ \iota)_M, M)$, defining a transformation $\xi : \lim \circ \iota \to \mathrm{id}_{\mathcal{C}}$. For $F \in \bar{\mathcal{C}}_0$, since $(\iota \circ \lim)_F \in [F]$, we can choose some $\zeta_F \in \bar{\mathcal{C}}((\iota \circ \lim)_F, F)$, defining a transformation $\zeta : \iota \circ \lim \to \mathrm{id}_{\bar{\mathcal{C}}}$. $\square$

The conditions of being continuous or Lipschitz can be considered for double functors between bidistortion functors as special cases of distortion functors (Section 10.4). Similarly, the relations of being equivalent or Lipschitz equivalent can be considered for bidistortions of the same involutive u-functor $\mathcal{C}/\mathcal{D}$ with a crossing on $\mathcal{D}$. If $F/G : (\mathcal{C}/\mathcal{D}, b) \to (\mathcal{C}'/\mathcal{D}', b')$ is a continuous (respectively, Lipschitz) double functor between bidistortion functors, then the induced map $\bar{F} : \mathcal{C}\bar{\mathcal{C}}_0 \to \mathcal{C}'\bar{\mathcal{C}}'_0$ is continuous (respectively, Lipschitz).

**Proposition 10.93.** *The property of being complete is invariant by Lipschitz equivalences of bidistortions.*

*Proof.* Lipschitz equivalent bidistortions have the same a-commutative a-functors and a-natural transformations. $\square$

## 10.11 The Hausdorff property

For $M, N \in \mathcal{C}_0$, let $D_b : \tilde{\mathcal{D}}(M, N) \times \tilde{\mathcal{D}}(M, N) \to [0, \infty]$ be the map defined by

$$D_b(\sigma, \tau) = \inf\{\max\{\{\sigma_1, \sigma_2\}^\times, \{\sigma_2, \sigma_3\}^\times, \dots, \{\sigma_{n-1}, \sigma_n\}^\times\} \mid (\sigma_1, \dots, \sigma_n) \in S(\sigma, \tau)\},$$

where $S(\sigma, \tau)$ is the set of finite sequences $(\sigma_1, \dots, \sigma_n)$ in $\tilde{\mathcal{D}}(M, N)$, with arbitrary length $n$, so that $\sigma_1 = \sigma$ and $\sigma_n = \tau$. From (b2), it follows that

$$(10.39) \qquad D_b(\sigma, \tau) \approx \{\sigma, \tau\}^\times$$

for $\sigma, \tau \in \tilde{\mathcal{D}}(M, N)$. The following result is elementary.

**Proposition 10.94.** *$D_b$ is a pseudo-ultrametric on $\tilde{\mathcal{D}}(M, N)$ with possible infinite values.*

From now on, $\tilde{\mathcal{D}}(M, N)$ will be endowed with $D_b$. Let $(\sigma_n)$ be a sequence in $\tilde{\mathcal{D}}(M, N)$. It is said that $(\sigma_n)$ is a-undistorted if $b(\sigma_n) \to 0$ as $n \to \infty$. According to (d1), this condition means that $\sigma_n$ “almost belong” to $\mathcal{C}(M, N)$ when $n \to \infty$, but in a sense weaker than $D_b(\sigma_n, \mathcal{C}(M, N)) \to 0$. So the hypothesis of the following result can be considered as some kind of compactness condition.
**Proposition 10.95.** If any a-undistorted sequence in $\tilde{D}(M, N)$ has some convergent subsequence for all $M, N \in \mathbb{C}_0$, then $\mathbb{C}\backslash\mathbb{C}_0$ is Hausdorff.

*Proof.* Suppose that $d_b(M, N) = 0$ for some $M, N \in \mathbb{C}_0$. By Lemma 10.84 and (b2), this means that there is an a-undistorted sequence $(\sigma_n)$ in $\tilde{D}(M, N)$. Then some subsequence $(\sigma_{n_m})$ of $(\sigma_n)$ converges to some $\sigma \in \tilde{D}(M, N)$. By (b4), (c3) and (10.39),

$$ \sigma \approx \{\sigma\}^{\times} \preceq \{\sigma, \sigma_{n_m}\}^{\times} \approx D_b(\sigma, \sigma_{n_m}), $$

which converges to zero as $n \to \infty$, yielding $b(\sigma) = 0$, and thus $\sigma \in \mathbb{C}(M, N)$ by (d1). So $[M] = [N]$ in $\mathbb{C}\backslash\mathbb{C}_0$. $\square$

Now, let $F, F' \in \bar{\mathbb{C}}_0$, and consider a sequence of transformations $t^n : A^n \to A'^n$ in $\Lambda_1(F, F')$. The source category of $A^n$ and $A'^n$ is denoted by $\mathcal{J}^n$. Let $\mathbf{a}^n : \mathcal{J}^n \to \mathcal{J}$ and $\mathbf{a}'^n : \mathcal{J}^n \to \mathcal{J}'$ be cofinal stp-reparametrizations defining $A^n$ and $A'^n$, respectively.

**Definition 10.96.** It is said that:

* $(t^n)$ is a-undistorted if $b^{\mathcal{J}^n}(t^n) \to 0$ as $n \to \infty$; and

* $(t^n)$ is Cauchy if it is a-undistorted and moreover

$$ \limsup_{j, k \to \infty} \bigcup_{f \in \mathcal{J}(\mathbf{a}_j^m, \mathbf{a}_k^n), f' \in \mathcal{J}'(\mathbf{a}'_j^m, \mathbf{a}'_k^n)} \{F'(f') t_j^m, t_k^n F(f)\}^\times $$

converges to zero as $m, n \to \infty$, where $(j, k)$ runs in the subset of $\mathcal{J}_0^m \times \mathcal{J}_0^n$ determined by the conditions $\mathbf{a}_j^m \le \mathbf{a}_k^n$ and $\mathbf{a}'_j^m \le \mathbf{a}'_k^n$.

In Definition 10.96, the a-undistortion condition means that $t^n$ is “almost a-natural” when $n \to \infty$, and thus it “almost represents” a morphism of $\bar{\mathbb{C}}(F, F')$. If $(t^n)$ is Cauchy, then $t^m$ and $t^n$ becomes “almost asymptotic” as $m, n \to \infty$ (see Definition 10.75). In fact, the supremum limit of that definition can be considered as a measure of how far are the “almost a-natural” transformations $t^m$ and $t^n$ from being asymptotic; i.e., it plays the role of a distance between the “virtual morphisms” of $\bar{\mathbb{C}}$ “almost represented” by $t^m$ and $t^n$. This justifies the term “Cauchy”. In this sense, the hypothesis of the following result can be considered as some kind of pre-compactness condition.

**Proposition 10.97.** If any a-undistorted sequence in $\Lambda_1(F, F')$ has some Cauchy subsequence for all $F, F' \in \bar{\mathbb{C}}_0$, then $\bar{\mathbb{C}}\backslash\bar{\mathbb{C}}_0$ is Hausdorff.

*Proof.* Suppose that $\bar{d}_b(F, F') = 0$ for some $F, F' \in \bar{\mathbb{C}}_0$, whose source categories are respectively denoted by $\mathcal{J}$ and $\mathcal{J}'$. Since $F$ and $F'$ are a-functors, for each $n \in \mathbb{N}$, there is some $i_n^* \in \mathcal{J}_0$ and some $i_n'^* \in \mathcal{J}'_0$ such that

$$ (10.40) \qquad i_n^* \le i_1, i_2, i_3 \implies \bigcup_{p \in \mathcal{J}(i_1, i_2), q \in \mathcal{J}(i_2, i_3)} \{F(qp), F(q)F(p)\}^\times < \frac{1}{2^n}, $$

$$ (10.41) \qquad i_n^{*} \le i'_1, i'_2, i'_3 \implies \bigcup_{p' \in \mathcal{J}'(i'_1, i'_2), q' \in \mathcal{J}'(i'_2, i'_3)} \{F'(q'p'), F'(q')F'(p')\}^{\times} < \frac{1}{2^n}. $$
By using (d2) with $\mathfrak{b}^\mathcal{J}$ for any small weakly filtered category $\mathcal{J}$, it follows from $\bar{\mathfrak{d}}_b(F, F') = 0$ that there is an $\mathfrak{a}$-undistorted sequence of transformations $t^n : A^n \to A'^n$ in $\Lambda_1(F, F')$. By taking a subsequence if necessary, we can assume that $\mathfrak{b}^{\mathcal{J}^n}(t^n) < 1/2^n$, where $\mathcal{J}^n$ is the source category of $A^n$ and $A'^n$. Then, for each $n \in \mathbb{N}$, there is some $j_n^* \in \mathcal{J}_0^n$ such that

$$ (10.42) \qquad j_n^* \le j, k \implies \bigcup_{f \in \mathcal{J}^n(j,k)} \{A^n(f) t_j^n, t_k^n A'^n(f)\}^\times < \frac{1}{2^n}. $$

Given cofinal stp-reparametrizations $\mathfrak{a}^n : \mathcal{J}^n \to \mathcal{J}$ and $\mathfrak{a}'^n : \mathcal{J}^n \to \mathcal{J}'$ defining $A^n$ and $A'^n$, respectively, we can assume that

$$ (10.43) \qquad \mathfrak{a}_{j_n^*}^n \le i_n^*, \qquad \mathfrak{a}_{j_n^*}^{n'} \le i_n^{*'} $$

for all $n \in \mathbb{N}$. Moreover $(t_n)$ has some Cauchy subsequence by hypothesis; indeed, we can assume that $(t_n)$ itself is Cauchy. Hence, by taking again a subsequence if necessary, we can also suppose that

$$ (10.44) \qquad \begin{aligned} & j_n^* \le j, \quad j_{n+1}^* \le k, \quad \mathfrak{a}_j^n \le \mathfrak{a}_k^{n+1}, \quad \mathfrak{a}_j'^n \le \mathfrak{a}_k'^{n+1} \\ & \implies \bigcup_{f \in \mathcal{J}(\mathfrak{a}_j^n, \mathfrak{a}_k^{n+1}), f' \in \mathcal{J}'(\mathfrak{a}_j'^n, \mathfrak{a}_k'^{n+1})} \{F'(f') t_j^n, t_k^{n+1} F(f)\}^\times < \frac{1}{2^n}, \end{aligned} $$

Let $\hat{\mathcal{J}}^n$ be the full subcategory of each $\mathcal{J}^n$ with $\hat{\mathcal{J}}_0^n = \{j \in \mathcal{J}_0^n \mid j_n^* \le j\}$, and let $\mathcal{K}$ be the small category described as follows:

* $\mathcal{K}_0 = \bigsqcup_{n=1}^\infty \hat{\mathcal{J}}_0^n = \bigsqcup_{n=1}^\infty (\{n\} \times \hat{\mathcal{J}}_0^n)$.

* For $(n, j), (n', j') \in \mathcal{K}_0$ with $n > n'$, we have $\mathcal{K}(n, j; n', j') = \emptyset$.

* For $(n, j), (n', j') \in \mathcal{K}_0$ with $n \le n'$, $\mathcal{K}(n, j; n', j')$ is the set of sequences

$$ (f_{n'}, f_{n'-1}, \dots, f_n) \in \hat{\mathcal{J}}_1^{n'} \times \hat{\mathcal{J}}_1^{n'-1} \times \dots \times \hat{\mathcal{J}}_1^n $$

such that, if $j_m$ and $k_m$ respectively denote the source and target of $f_m$ for $m \in \{n, n+1, \dots, n'\}$, then $k_{n'} = j'$, $j_n = j$, and

$$ \mathfrak{a}_{j_m}^m \ge \mathfrak{a}_{k_{m-1}}^{m-1}, \qquad \mathfrak{a}_{j_m}^{m'} \ge \mathfrak{a}_{k_{m-1}}^{m'-1} $$

if $m > n$.

* For $(n, j), (n', j'), (n'', j'') \in \mathcal{K}_0$ with $n \le n' \le n''$, and

$$ \lambda = (f_{n'}, f_{n'-1}, \dots, f_n) \in \mathcal{K}(n, j; n', j'), $$

$$ \mu = (g_{n''}, g_{n''-1}, \dots, g_{n'}) \in \mathcal{K}(n', j'; n'', j''), $$

define

$$ \mu\lambda = (g_{n''}, g_{n''-1}, \dots, g_{n'}f_{n'}, f_{n'-1}, \dots, f_n) \in \mathcal{K}(n, j; n'', j''). $$
On the other hand, by Lemma 10.71, (10.40), (10.41) and (10.43),

$$
\begin{align*}
\{B(\lambda), \sigma\}^\times &\asymp \{B(\lambda), F(\mathbf{a}(f_{n'})) h_{k_{n'-1}, j_{n'}} \mathbf{a}(f_{n'-1}) \cdots \mathbf{a}(f_{n+1}) h_{k_n, j_{n+1}}\} A(f_n)^\times \\
&\phantom{\asymp} \cup \{F(\mathbf{a}(f_{n'})) h_{k_{n'-1}, j_{n'}} \mathbf{a}(f_{n'-1}) \cdots \mathbf{a}(f_{n+1}) h_{k_n, j_{n+1}}, \\
&\phantom{\asymp} \qquad F(\mathbf{a}(f_{n'})) h_{k_{n'-1}, j_{n'}} \mathbf{a}(f_{n'-1}) \cdots \mathbf{a}(f_{n+1}) F(h_{k_n, j_{n+1}})\}^\times \cup \cdots \\
&\phantom{\asymp} \cup \{F(\mathbf{a}(f_{n'})) h_{k_{n'-1}, j_{n'}} \mathbf{a}(f_{n'-1}), F(\mathbf{a}(f_{n'})) h_{k_{n'-1}, j_{n'}} A(f_{n'-1})\}^\times \\
&\phantom{\asymp} \cup \{F(\mathbf{a}(f_{n'})) h_{k_{n'-1}, j_{n'}} F(\mathbf{a}(f_{n'})) F(h_{k_{n'-1}, j_{n'}})\}^\times \\
&\phantom{\asymp} \cup \{A(f_n) + F(h_{k_n, j_{n+1}}) + A(f_{n+1}) + \cdots + F(h_{k_{n'-2}, j_{n'-1}}) + A(f_{n'-1})\} \\
&\asymp \frac{1}{2^n} + \cdots + \frac{1}{2^{n'-1}}.
\end{align*}
$$

Similarly,

$$
\{B'(\lambda), \sigma'\}^\times \asymp \frac{1}{2^n} + \cdots + \frac{1}{2^{n'-1}}.
$$

Therefore, by (b2), Lemma 10.64 and (10.42),

$$
\begin{align*}
\{u_{n',j'} B(\lambda), B'(\lambda) u_{n,j}\}^\times &\asymp \{t_{j'} B(\lambda), t_{j'} \sigma\}^\times \cup \{t_{j'} \sigma, \sigma' t_j\}^\times \cup \{\sigma' t_j, B'(\lambda) t_j\}^\times \\
&\asymp \{B(\lambda), \sigma\}^\times \cup \{t_{j'} \sigma, \sigma' t_j\}^\times \cup \{\sigma', B'(\lambda)\}^\times \cup \{t_j, t_{j'}\} \\
&\asymp \frac{1}{2^n} + \cdots + \frac{1}{2^{n'}},
\end{align*}
$$

which converges to zero as $n, n' \to \infty$ uniformly on $j, j'$ and $\lambda$. So $u$ is a-natural, and thus
represents a morphism $[u] : F \to F'$ in $\bar{\mathcal{C}}$. Hence $[F] = [F']$ in $\bar{\mathcal{C}} \setminus \bar{\mathcal{C}}_0$. $\square$

**Definition 10.98.** It is said that $\times$ is *asymptotically trivial*, or simply *a-trivial*, with respect to $\mathfrak{b}$ if $S \approx S^\times$ for $S \in \mathcal{P}_{\tilde{D},1}$.

Of course, trivial crossings are a-trivial with respect to any bidistortion. The proof of the
following result is elementary.

**Proposition 10.99.** If $\times$ is a-trivial, then:

* any Cauchy weakly filtered stp-map to $\tilde{D}$ is an a-commutative a-functor;

* any a-undistorted transformation between Cauchy weakly filtered stp-maps to $\tilde{D}$ is a-natural;

* any pair of transformations in $\Lambda_1(F, F')$ are asymptotic for all $F, F' \in \bar{C}_0$; and

* any a-undistorted sequence in $\Lambda_1(F, F')$ is Cauchy for all $F, F' \in \bar{C}_0$.

**Corollary 10.100.** If $\times$ is a-trivial, then $\bar{C}$ is the groupoid defined by an equivalence relation, and $\bar{C}\backslash\bar{C}_0$ is Hausdorff.

*Proof.* The first assertion holds because $\bar{C}$ is a groupoid (Proposition 10.83) and $\bar{C}(F, F')$ has at most one element for all $F, F' \in \bar{C}_0$ (Proposition 10.99). The second assertion is a direct consequence of Propositions 10.97 and 10.99. $\square$
## 10.12 Examples of bidistortions

**Example 10.101 (Bidistortions of $\mathcal{C}/\mathcal{B}^{\text{in}}$).** For a category $\mathcal{B}$, consider its involutive category $\mathcal{B}^{\text{in}}$ with its canonical crossing (Example 10.57). Then, given any involutive u-functor $\mathcal{C}/\mathcal{B}^{\text{in}}$, a distortion $\mathfrak{b}$ of $\mathcal{C}/\mathcal{B}^{\text{in}}$ is bidistortion if and only if the following properties are satisfied:

(b1') $\mathfrak{b}(f, g) \approx \mathfrak{b}(g, f)$ for morphisms $(f, g)$ of $\widetilde{\mathcal{B}}^{\text{in}}$.

(b2') For finite sequences $((f_1, g_1), \dots, (f_n, g_n))$ of parallel morphisms of $\widetilde{\mathcal{B}}^{\text{in}}$ with arbitrary length $n$,

$$ \mathfrak{b}(f_1, g_n) \preccurlyeq \sup \mathfrak{b}(\{(f_1, g_1), (g_1, f_2), (f_2, g_2), \dots, (g_{n-1}, f_n), (f_n, g_n)\}). $$

(b3') For morphisms $(f, g)$ of $\widetilde{\mathcal{B}}^{\text{in}}$,

$$ \mathfrak{b}(gf, \text{id}) \preccurlyeq \mathfrak{b}(f, g). $$

This can be seen as follows. Suppose that $\mathfrak{b}$ satisfies (b1')-(b3'). Properties (b4)-(b6) hold because $\times$ is a functorial pairing. Since (b1') is (10.31) in this example, (b1) follows because $\times$ is involutive. Property (b3') easily yields (b3) by (d1) and (d2). Property (b2) follows easily from (b2').

Now, assume that $\mathfrak{b}$ satisfies (b1)-(b6). Then (b1') and (b3') are respectively given by (10.31) and (b3). Finally, (b2') for sequences $(\sigma_1, \dots, \sigma_n)$ follows by applying (b2) to the corresponding sequences $(\sigma_1, \tau_1, \sigma_2, \dots, \sigma_{n-1}, \tau_{n-1}, \sigma_n)$ and using (b1'), where $\tau_j = (f_{j+1}, g_j)$ if $\sigma_i = (f_i, g_i)$.

**Example 10.102 (Bidistortions of $\text{Iso}(\mathcal{A})/\mathcal{B}^{\text{in}}$).** As a particular case of Example 10.101, consider a bidistortion $\mathfrak{b}$ of the involutive u-functor $\text{Iso}(\mathcal{A})/\mathcal{B}^{\text{in}}$ induced by a u-functor $\mathcal{A}/\mathcal{B}$ (Example 10.13). For any morphism $(f, g)$ of $\widetilde{\mathcal{B}}^{\text{in}}$, if $\mathfrak{b}(f, g)$ is small, then, somehow, $f$ and $g$ almost are isomorphisms of $\mathcal{A}$ and inverse of each other, written $g \approx f^{-1}$ or $f \approx g^{-1}$. So, given another morphism $(f', g')$ of $\widetilde{\mathcal{B}}^{\text{in}}$ parallel to $(f, g)$, if $\mathfrak{b}(f, g)$ and $\mathfrak{b}(f', g')$ are small, then $\mathfrak{b}(f, g')$ gives an "approximate distance" between $(f, g)$ and $(f', g')$; indeed, if $\mathfrak{b}(f, g')$ is also small, then $f \approx g'^{-1} \approx f'$ and $g \approx f^{-1} \approx g'$. In this sense, if $\sup \mathfrak{b}(S)$ is small for some set $S$ of parallel morphisms of $\widetilde{\mathcal{B}}^{\text{in}}$, then $\sup \mathfrak{b}(S^{\times})$ really plays the role of "approximate diameter" of $S$, which was the key idea of Sections 10.7–10.11.

**Example 10.103 (Bidistortions of $\mathcal{A}/\mathcal{A}^{\text{in}}$).** Another particular case of Example 10.101 is a bidistortion $\mathfrak{b}$ of the involutive u-functor $\mathcal{A}/\mathcal{A}^{\text{in}}$, given by the canonical injection of an involutive category $\mathcal{A}$ into $\mathcal{A}^{\text{in}}$ (Example 10.8). For a morphism $(f, g)$ of $\mathcal{A}^{\text{in}}$, the number $\mathfrak{b}(f, g)$ measures how far is $g$ from being $\bar{f}$. So, like in Example 10.102, if $\sup \mathfrak{b}(S)$ is small for some set $S$ of parallel morphisms of $\mathcal{A}^{\text{in}}$, then $\sup \mathfrak{b}(S^{\times})$ can be really understood as an "approximate diameter" of $S$.

Some terminology of Sections 10.8–10.11 is motivated by the following example.
**Example 10.104 (Top bidistortion).** For any involutive u-functor $\mathcal{C}/\mathcal{D}$, where $\mathcal{D}$ is endowed with a crossing, the top distortion of $\mathcal{C}/\mathcal{D}$ is a bidistortion called the *top bidistortion*. It induces the top metric with possible infinite values *** on $\mathcal{C}\backslash\mathcal{C}_0$.

**Example 10.105 (Pseudo-metrics are bidistortions).** For any set $X \subset \mathcal{U}$, let $\mathcal{C}$ be the groupoid defined by any equivalence relation $\mathcal{R}$ on $X$, and let $\mathcal{D}$ be the groupoid defined by the equivalence relation $X \times X$ on $X$. Since

$$ (10.45) \qquad \mathcal{D}(x, y) = \{(x, y)\} $$

for all $x, y \in X$, $\mathcal{D}$ can only be endowed with the trivial crossing, which indeed corresponds to the canonical crossing on $\mathcal{D}^{\text{in}}$ by the identity $\mathcal{D} \equiv \mathcal{D}^{\text{in}}$ given by $(x, y) \equiv ((x, y), (y, x))$; in particular, this is another special case of Example 10.101. The bidistortions of $\mathcal{C}/\mathcal{D}$ are its “almost involutive” distortions; i.e., the maps $\mathfrak{b}: \mathcal{D}_1 = X \times X \to [0, \infty]$ satisfying

$$ \begin{aligned} \mathfrak{b}(x, y) &\approx \mathfrak{b}(y, x), \\ \mathfrak{b}(x, y) = 0 &\iff (x, y) \in \mathcal{R}, \\ \mathfrak{b}(x_1, x_n) &\le \mathfrak{b}(x_1, x_2) + \cdots + \mathfrak{b}(x_{n-1}, x_n); \end{aligned} $$

in fact, this properties become (b1'), (d1) and (d2) in this case, and the conditions (b2') and (b3') hold by (10.45).

In particular, any pseudo-metric $d$ on $X$ is a bidistortion of $\mathcal{C}/\mathcal{D}$ for $\mathcal{R} = d^{-1}(0)$. In this case, consider the induced metric on $X/\mathcal{R}$, and let $\bar{X}/\mathcal{R}$ denote its completion. Observe that $\mathcal{C}\backslash\mathcal{C}_0 = X/\mathcal{R}$. By (10.45), any stp-map to $\mathcal{D}$ is a commutative functor, and any transformation between functors to $\mathcal{D}$ is natural. Then a weakly filtered stp-map to $\mathcal{D}$ is an a-commutative a-functor if and only if it is Cauchy, which in turn holds when the corresponding net in $X$ (Section 10.1) is Cauchy, which means that its projection to $X/\mathcal{R}$ is a Cauchy net. So, by taking limits in $\bar{X}/\mathcal{R}$ of nets in $X/\mathcal{R}$, we get a canonical category equivalence between $\bar{\mathcal{C}}$ and the groupoid defined by the trivial equivalence relation on $\bar{X}/\mathcal{R}$. Thus $\bar{\mathcal{C}}\backslash\bar{\mathcal{C}}_0$ is isometric to $\bar{X}/\mathcal{R}$, and $\bar{\iota}: \mathcal{C}\backslash\mathcal{C}_0 \to \bar{\mathcal{C}}\backslash\bar{\mathcal{C}}_0$ corresponds to the canonical injection $X/\mathcal{R} \to \bar{X}/\mathcal{R}$. Moreover $d$ is complete as bidistortion if and only if it is complete as pseudo-metric.

**Example 10.106 (Bidistortion of $\mathcal{A}^{\text{in}}/\mathcal{B}^{\text{in}}$ induced by a distortion of $\mathcal{A}/\mathcal{B}$).** For a u-functor $\mathcal{A}/\mathcal{B}$, consider the induced involutive u-functor $\mathcal{A}^{\text{in}}/\mathcal{B}^{\text{in}}$ (Example 10.12), and the corresponding lift $\widetilde{\mathcal{B}}^{\text{in}}$ of $\mathcal{B}^{\text{in}}$. Given a distortion $\mathfrak{d}$ of $\mathcal{A}/\mathcal{B}$, let $\mathfrak{b}: \widetilde{\mathcal{B}}^{\text{in}} \to [0, \infty]$ be defined by

$$ \mathfrak{b}(f, g) = \|(\mathfrak{d}(f), \mathfrak{d}(g))\|, $$

where $\| \|$ is any norm on $\mathbb{R}^2$. Let us prove that this map is a bidistortion of $\mathcal{A}^{\text{in}}/\mathcal{B}^{\text{in}}$. First, (b1') and (b2') are evident for $\mathfrak{b}$, and (b3') for $\mathfrak{b}$ follows from (d2) for $\mathfrak{d}$. Moreover (d1) for $\mathfrak{b}$ is an easy consequence of (d1) for $\mathfrak{d}$. Thus, according to Example 10.101, it only remains to prove that $\mathfrak{b}$ satisfies (d2). For that purpose, fix some $C, D \ge 1$ such that

$$ \frac{\max\{x, y\}}{C} \le \|(x, y)\| \le D \max\{x, y\}. $$
and $\mathfrak{b}'$ be bidistortions of $\mathcal{C}/\mathcal{D}$ and $\mathcal{C}'/\mathcal{D}'$, respectively, and let $\mathfrak{b}''$ be their fibered product distortion (Example 10.44). By using (10.13), (10.28) and (10.29), it follows easily that $\mathfrak{b}''$ is a bidistortion of $\mathcal{C}''/\mathcal{D}''$.

Since

$$ \tilde{\mathcal{D}}'' \equiv \tilde{\mathcal{D}} \times_{\tilde{\mathfrak{g}}} \tilde{\mathcal{D}}' \subset \tilde{\mathcal{D}} \times \tilde{\mathcal{D}}' $$

by (10.7), each stp-map with target $\tilde{\mathcal{D}}''$ can be considered as a pair of stp-maps with targets $\tilde{\mathcal{D}}$ and $\tilde{\mathcal{D}}'$. Similarly, any transformation between functors to $\tilde{\mathcal{D}}''$ can be considered as a pair of transformations between functors to $\tilde{\mathcal{D}}$ and $\tilde{\mathcal{D}}'$. From the definition of $\mathfrak{b}'$, it follows that:

* a weakly filtered stp-map to $\tilde{\mathcal{D}}''$ is an a-commutative a-functor if and only if its components are a-commutative a-functors;

* a transformation between weakly filtered stp-maps to $\tilde{\mathcal{D}}''$ is a-natural if and only if its components are a-natural; and

* two transformations defining parallel morphisms of $\overline{\mathcal{C}''}$ are asymptotic if and only if their components are asymptotic.

So $\overline{\mathcal{C}''}$ can be canonically identified to a subgroupoid of $\overline{\mathcal{C}} \times \overline{\mathcal{C}}'$; precisely,

$$ \overline{\mathcal{C}''_0} = \{(F, F') \in \overline{\mathcal{C}}_0 \times \overline{\mathcal{C}}'_0 \mid \tilde{Q}F = \tilde{Q}'F'\} \subset \overline{\mathcal{C}}_0 \times \overline{\mathcal{C}}'_0, $$

and

$$ \begin{align*} \overline{\mathcal{C}''}(M, M'; N, N') &\equiv \{[(t, t')] \mid t \in \Lambda_1(F, G), t' \in \Lambda_1(F', G'), \tilde{Q}t = \tilde{Q}'t'\} \\ &\equiv \{([t], [t']) \mid t \in \Lambda_1(F, G), t' \in \Lambda_1(F', G'), \tilde{Q}t = \tilde{Q}'t'\} \\ &\subset \overline{\mathcal{C}}(F, G) \times \overline{\mathcal{C}}'(F', G') \end{align*} $$

for $(F, F')$, $(G, G') \in \overline{\mathcal{C}}''_0$. This the canonical functor $\iota'' : \mathcal{C}'' \to \overline{\mathcal{C}''}$ can be identified to the restriction of the product of canonical functors $\iota \times \iota' : \mathcal{C} \times \mathcal{C}' \to \overline{\mathcal{C}} \times \overline{\mathcal{C}}'$.

Now, consider the notation of Example 10.44, and suppose that (10.46) is the diagram

$$ \begin{tikzcd} \mathrm{Iso}(\mathcal{A}) \arrow[r, "P"] & \mathrm{Iso}(\mathcal{C}) \arrow[r, "P'"] & \mathrm{Iso}(\mathcal{A}') \\ \llap{$|$} \mathrm{iso/in} \arrow[d] & \llap{$|$} \mathrm{iso/in} \arrow[d] & \llap{$|$} \mathrm{iso/in} \arrow[d] \\ \mathcal{B}^{\mathrm{in}} \arrow[r, "Q^{\mathrm{in}}"] & \mathcal{D}^{\mathrm{in}} \arrow[r, "Q'^{\mathrm{in}}"] & \mathcal{B}'^{\mathrm{in}} \end{tikzcd} $$

induced by (10.18). Then

$$ (\mathrm{Iso}(\mathcal{A}) \times_{\mathrm{Iso}(\mathcal{C})} \mathrm{Iso}(\mathcal{A}'))/(\mathcal{B}^{\mathrm{in}} \times_{\mathcal{D}^{\mathrm{in}}} \mathcal{B}'^{\mathrm{in}}) \equiv \mathrm{Iso}(\mathcal{A}'')/\mathcal{B}''^{\mathrm{in}} $$

by (10.4) and (10.8). Moreover, by this identity, the fibered products of the canonical involutions and crossings of $\mathrm{Iso}(\mathcal{A})/\mathcal{B}^{\mathrm{in}}$ and $\mathrm{Iso}(\mathcal{A}')/\mathcal{B}'^{\mathrm{in}}$ correspond to the canonical involution and crossing of $\mathrm{Iso}(\mathcal{A}'')/\mathcal{B}''^{\mathrm{in}}$ (Examples 10.3 and 10.58). Thus $\mathfrak{b}''$ can be considered as a bidistortion of $\mathrm{Iso}(\mathcal{A}'')/\mathcal{B}''^{\mathrm{in}}$.
**Example 10.109.** Following Example 10.43, any bidistortion induces an involutive bidistortion.

**Example 10.110.** For a u-functor $\mathcal{A}/\mathcal{B}$, let $\mathfrak{d}$ be a distortion of $\text{Iso}(\mathcal{A})/\text{Iso}(\mathcal{B})$ satisfying

$$ (10.47) \qquad \mathfrak{d}(f) \approx \mathfrak{d}(f^{-1}) $$

for $f \in \widetilde{\text{Iso}(\mathcal{B})}_1$. We can use Example 10.43 to achieve (10.47) when $\mathfrak{d}$ is arbitrary. Then a bidistortion $\mathfrak{b}$ of $\text{Iso}(\mathcal{A})/\mathcal{B}^{\text{in}}$ is defined by

$$ \mathfrak{b}(f,g) = \begin{cases} \mathfrak{d}(f) & \text{if } f \in \widetilde{\text{Iso}(\mathcal{B})}_1 \text{ and } g = f^{-1} \\ \infty & \text{otherwise.} \end{cases} $$

Observe that

$$ (10.48) \qquad \mathfrak{b}(f,g) < \infty \implies f,g \in \widetilde{\text{Iso}(\mathcal{B})}_1 \text{ and } g = f^{-1}. $$

Vice versa, any bidistortion $\mathfrak{b}$ of $\text{Iso}(\mathcal{A})/\mathcal{B}^{\text{in}}$ induces an involutive distortion $\mathfrak{d}$ of $\text{Iso}(\mathcal{A})/\text{Iso}(\mathcal{B})$ defined by $\mathfrak{d}(f) = \mathfrak{b}(f, f^{-1})$. This defines a bijection between the family of distortions of $\text{Iso}(\mathcal{A})/\text{Iso}(\mathcal{B})$ satisfying (10.47) and the family of bidistortions $\mathfrak{b}$ of $\text{Iso}(\mathcal{A})/\mathcal{B}^{\text{in}}$ satisfying (10.48). Moreover involutive bidistortions correspond to involutive distortions.

**Example 10.111.** Let $\mathcal{A}/\mathcal{B}$ be a u-functor over Set. For a good class $\Xi$ of sub-objects of $\mathcal{B}$, let $\text{Par}$ and $\text{ParIso}$ denote the corresponding categoris of $\Xi$-partial morphisms and $\Xi$-partial isomorphisms, respectively (Example 10.21); recall that $\text{ParIso}$ is involutive. Let $\mathfrak{d}$ be a distortion of $\text{Iso}(\mathcal{A})/\text{ParIso}$ satisfying the corresponding version of (10.47) for $f \in \widetilde{\text{ParIso}}_1$, which is not a big restriction by Example 10.43. It induces a bidistortion $\mathfrak{b}$ of $\text{Iso}(\mathcal{A})/\text{Par}^{\text{in}}$ by setting

$$ \mathfrak{b}(f,g) = \begin{cases} \mathfrak{d}(f) & \text{if } f,g \in \widetilde{\text{ParIso}}_1 \text{ and } g = f^{-1} \\ \infty & \text{otherwise,} \end{cases} $$

which satisfies the version of (10.48) with $\text{ParIso}$ instead of $\text{Iso}$. Like in Example 10.110, this procedure establishes a bijection between the families of distortions of $\text{Iso}(\mathcal{A})/\text{ParIso}$ and bidistortions of $\text{Iso}(\mathcal{A})/\text{Par}^{\text{in}}$ satisfying the above conditions.

**Example 10.112.** Let $\mathcal{A}/\mathcal{B}$ be a u-functor over $\text{Met}/\text{ParSet}$ or $\text{Path}/\text{ParSet}$. For $M \in \mathcal{A}_0$ and $f \in \widetilde{\mathcal{B}}(M, M)$, let

$$ D_M(f) = \sup\{d_M(x, f(x)) \mid x \in \text{dom } f\}, $$

where $d_M$ denotes the underlying metric of $M$. Any distortion $\mathfrak{d}$ of $\mathcal{A}/\mathcal{B}$ induces a bidistortion $\mathfrak{b}$ of $\text{Iso}(\mathcal{A})/\mathcal{B}^{\text{in}}$ defined by

$$ \mathfrak{b}(f,g) = \|(\mathfrak{d}(f), \mathfrak{d}(g), D_M(g \circ f), D_N(f \circ g))\| $$

for $M, N \in \mathcal{A}_0$ and $(f, g) \in \widetilde{\mathcal{B}}^{\text{in}}(M, N)$, where $\| \|$ is any fixed norm on $\mathbb{R}^4$. If this norm is invariant by permutations of the first two components and last two components, then $\mathfrak{b}$ is involutive.
**Example 10.113 (The Lipschitz bidistortion).** With the notation of Example 10.45 and according to Example 10.112, $\mathfrak{d}_{\text{Lip}}$ induces a bidistortion $\mathfrak{b}_{\text{Lip}}$ of Iso(Met)/Set<sup>in</sup> called the Lipschitz bidistortion.

**Example 10.114 (The large scale Lipschitz bidistortion).** With the notation of Example 10.46 and according to Example 10.112, $\mathfrak{d}_{\text{lsl}}$ induces a bidistortion $\mathfrak{b}_{\text{lsl}}$ of Iso(Met)/Set<sup>in</sup> called the large scale Lipschitz bidistortion. By ***, the restriction of the corresponding distance $d_{\mathfrak{b}_{\text{lsl}}}$ to $\mathcal{M}_c = \text{Iso}(\text{Met}_c)\setminus\text{Met}_{c,0}$ is Lipschitz equivalent to the Gromov-Hausdorff distance, and thus induces the Gromov topology.

**Example 10.115 (The arc-wise Lipschitz bidistortion).** With the notation of Example 10.47, and following Example 10.112, $\mathfrak{d}_{\text{aLip}}$ induces a bidistortion $\mathfrak{b}_{\text{aLip}}$ of Iso(Path)/Set<sup>in</sup> called the arc-wise Lipschitz bidistortion.

**Example 10.116 (The arc-wise large scale Lipschitz bidistortion).** With the notation of example 10.48 and using Example 10.112, $\mathfrak{d}_{\text{alsl}}$ induces a bidistortion $\mathfrak{b}_{\text{alsl}}$ of Iso(Path)/Set<sup>in</sup>, called the arc-wise large scale Lipschitz bidistortion.

**Example 10.117.** With the notation of Examples 10.50 and 10.51, and according to Example 10.112, $\mathfrak{d}^*$ and $\mathfrak{d}^\dagger$ induce bidistortions $\mathfrak{b}^*$ and $\mathfrak{b}^\dagger$ of Iso($\mathcal{A}_*$)/$\mathcal{B}^\text{in}$. For instance, $\mathfrak{d}^*_{\text{Lip}}$, $\mathfrak{d}^*_{\text{lsl}}$, $\mathfrak{d}^\dagger_{\text{Lip}}$ and $\mathfrak{d}^\dagger_{\text{lsl}}$ induce bidistortions $\mathfrak{b}^*_{\text{Lip}}$, $\mathfrak{b}^*_{\text{lsl}}$, $\mathfrak{b}^\dagger_{\text{Lip}}$ and $\mathfrak{b}^\dagger_{\text{lsl}}$ of Iso(Met<sub>*</sub>)/Set<sup>in</sup>; and $\mathfrak{d}^*_{\text{aLip}}$, $\mathfrak{d}^*_{\text{alsl}}$, $\mathfrak{d}^\dagger_{\text{aLip}}$ and $\mathfrak{d}^\dagger_{\text{alsl}}$ induce bidistortions $\mathfrak{b}^*_{\text{aLip}}$, $\mathfrak{b}^*_{\text{alsl}}$, $\mathfrak{b}^\dagger_{\text{aLip}}$ and $\mathfrak{b}^\dagger_{\text{alsl}}$ of Iso(Path<sub>*</sub>)/Set<sup>in</sup>. By ***, the restrictions of the metrics $d_{\mathfrak{b}^*_{\text{lsl}}}$ and $d_{\mathfrak{b}^\dagger_{\text{lsl}}}$ to $\mathcal{M}_{c,*} = \text{Iso}(\text{Met}_{c,*})\setminus\text{Met}_{c,*0}$ are Lipschitz equivalent to the pointed Gromov-Hausdorff distance.

**Example 10.118.** With the notation of Example 10.53, $\mathfrak{d}^{*\ddagger}$ and $\mathfrak{d}^{\dagger\ddagger}$ induce bidistortions $\mathfrak{b}^{*\ddagger}$ and $\mathfrak{b}^{\dagger\ddagger}$ of Iso($\mathcal{A}_*$)/Par($\mathcal{B}, \Xi$)<sup>in</sup> according to Example 10.112. For instance, $\mathfrak{d}^{*\ddagger}_{\text{Lip}}$, $\mathfrak{d}^{*\ddagger}_{\text{lsl}}$, $\mathfrak{d}^{\dagger\ddagger}_{\text{Lip}}$ and $\mathfrak{d}^{\dagger\ddagger}_{\text{lsl}}$ induce bidistortions $\mathfrak{b}^{*\ddagger}_{\text{Lip}}$, $\mathfrak{b}^{*\ddagger}_{\text{lsl}}$, $\mathfrak{b}^{\dagger\ddagger}_{\text{Lip}}$ and $\mathfrak{b}^{\dagger\ddagger}_{\text{lsl}}$ of Iso(Met<sub>*</sub>)/ParSet<sup>in</sup>; and $\mathfrak{d}^{*\ddagger}_{\text{aLip}}$ and $\mathfrak{d}^{\dagger\ddagger}_{\text{aLip}}$ induce bidistortions $\mathfrak{b}^{*\ddagger}_{\text{aLip}}$ and $\mathfrak{b}^{\dagger\ddagger}_{\text{aLip}}$ of Iso(Path<sub>*</sub>)/ParSet<sup>in</sup>. The restrictions of the metrics $d_{\mathfrak{b}^{*\ddagger}_{\text{lsl}}}$ and $d_{\mathfrak{b}^{\dagger\ddagger}_{\text{lsl}}}$ to $\mathcal{M}_* = \text{Iso}(\text{PropMet}_*)\setminus\text{PropMet}_{*,0}$ induce the pointed Gromov topology.

**Example 10.119.** For $C^\infty$ maps between connected Riemannian manifolds, $f, g: M \to N$, consider their tangent maps of order $m \in \mathbb{N}$, $T^m f, T^m g: T^m M \to T^m N$, and the induced metric tensors on $T^m M$ and $T^m N$ defined by iterative consideration of Sasakian metrics. Since $T^m M$ and $T^m M$ are connected, they are length spaces. Then set

$$D_{M,N}^{m}(f,g) = \inf\{R \ge 0 \mid T^m f \text{ and } T^m g \text{ are } R\text{-parallel}\}$$

if $T^m f$ and $T^m g$ are parallel, and $D_{M,N}^m(f,g) = \infty$ otherwise. The sequence $(D_{M,N}^m(f,g))$ will be denoted by $D_{M,N}(f,g)$. For a $C^\infty$ local map of $M$ to $M$, $f: U \to M$, let $D_M(f) = D_{U,M}(f, \iota_U)$, where $\iota_U$ denotes the inclusion map $U \hookrightarrow M$. Now, any distortion $\mathfrak{d}$ of a u-functor $\mathcal{A}/\mathcal{B}$ over ConRiem/LocMan<sup>\infty</sup> induces a bidistortion $\mathfrak{b}$ of $\mathcal{A}/\mathcal{B}$ as follows. For $M, N \in \mathcal{A}_0$ and $(f,g) \in \tilde{\mathcal{B}}^\text{in}(M, N)$, let

$$\mathfrak{b}_{M,N}(f,g) = \|(\mathfrak{d}_{M,N}(f), \mathfrak{d}_{M,N}(g), D_M(g \circ f), D_N(f \circ g))\|$$
if $\mathfrak{d}_{M,N}(f)$ and $\mathfrak{d}_{M,N}(g)$ are finite, and $D_M(g \circ f)$ and $D_N(f \circ g)$ are in $\ell^\infty$; and let $\mathfrak{b}_{M,N}(f,g) = \infty$ otherwise. Here, $\| \|$ is any fixed norm on $\mathbb{R}^2 \times (\ell^\infty)^2 = \ell^\infty(\{1,2\} \cup \mathbb{N} \cup \mathbb{N})$ invariant by permutations of the two components in $\mathbb{R}^2$ and of the two components in $(\ell^\infty)^2$, and satisfying the obvious property analogous to (10.20).

**Example 10.120** (The Riemannian bidistortion). With the notation of Example 10.49 and applying Example 10.119, $\mathfrak{d}_{\text{Riem}}$ induces a bidistortion $\mathfrak{b}_{\text{Riem}}$ of $\text{Iso}(\text{ConRiem})/\text{Man}^{\infty,\text{in}}$ called the *Riemannian bidistortion*. For each $n \in \mathbb{N}$, the restriction of the metric $d_{\mathfrak{b}_{\text{Riem}}}$ to

$$ \mathcal{M}_c(n) = \text{Iso}(\text{ConRiem}_c(n)) \setminus \text{ConRiem}_{c,0}(n) $$

induces the $C^\infty$ topology.

**Example 10.121** (The pointed Riemannian bidistortions). With the notation of Examples 10.50 and 10.51, and according to Example 10.119, $\mathfrak{d}_{\text{Riem}}^*$ and $\mathfrak{d}_{\text{Riem}}^\dagger$ induce a bidistortions $\mathfrak{b}_{\text{Riem}}^*$ and $\mathfrak{b}_{\text{Riem}}^\dagger$ of $\text{Iso}(\text{ConRiem}_*)/\text{Man}^{\infty,\text{in}}$ called the *Riemannian bidistortions*. For each $n \in \mathbb{N}$, the restriction of both metrics $d_{\mathfrak{b}_{\text{Riem}}^*}$ and $d_{\mathfrak{b}_{\text{Riem}}^\dagger}$ to

$$ \mathcal{M}_{c,*}(n) = \text{Iso}(\text{ConRiem}_{c,*}(n)) \setminus \text{ConRiem}_{c,*0}(n) $$

induce the pointed $C^\infty$ topology.

**Example 10.122.** With the notation of Example 10.53 and using Example 10.119, $\mathfrak{d}_{\text{Riem}}^{*\ddagger}$ and $\mathfrak{d}_{\text{Riem}}^{\ddagger\ddagger}$ induce bidistortions $\mathfrak{b}_{\text{Riem}}^{*\ddagger}$ and $\mathfrak{b}_{\text{Riem}}^{\ddagger\ddagger}$ of $\text{ConRiem}_* / \text{LocMan}^\infty$. For each $n \in \mathbb{N}$, the restrictions of both metrics $d_{\mathfrak{b}_{\text{Riem}}^{*\ddagger}}$ and $d_{\mathfrak{b}_{\text{Riem}}^{\ddagger\ddagger}}$ to

$$ \mathcal{M}_*(n) = \text{Iso}(\text{ComConRiem}_*(n)) \setminus \text{ComConRiem}_{*,0}(n) $$

induce the pointed $C^\infty$ topology.

In Examples 10.120–10.122, different bidistortions can be obtained by uning Example 10.112 instead of Example 10.119.

It seems that many other known mathematical constructions also fit into this framework. But we shall concentrate on its application to construct universal foliated spaces and study the generic properties of leaves.

## 10.13 Partitioned sets and spaces

A set endowed with an equivalence relation is called a *partitioned set*. A *partitioned map* is a map between partitioned sets compatible with the equivalence relations. Partitioned maps between small partitioned sets form a Set-category.

Let $\mathcal{A}$ be a category over ParSet. For each $M \in \mathcal{A}_0$, there is a canonical map $\iota_M : M \to \text{Iso}(\mathcal{A}_*) \setminus \mathcal{A}_{*,0}$ defined by $\iota_M(x) = [M, x]$, where $[M, x]$ denotes the class of $(M, x) \in \mathcal{A}_{*,0}$. Observe that this map is injective if and only if $M$ is aperiodic. The map $\iota_M$ induces an
injection $\bar{\iota}_M : \text{Iso}_A(M)\backslash M \to \text{Iso}(A_*)\backslash A_{*,0}$. The notation $\iota$ and $\bar{\iota}$ is also used when there is no confusion. The images of the maps $\iota_M$, with $M$ running in $A_0$, form a canonical partition of $\text{Iso}(A_*)\backslash A_{*,0}$, and the corresponding quotient set can be identified to $\text{Iso}(A)\backslash A_0$ via the canonical u-functor $A_*/A$. For any ParSet-functor $F : A \to B$, the corresponding map $F_{*,\text{iso}} : \text{Iso}(A_*)\backslash A_{*,0} \to \text{Iso}(B_*)\backslash B_{*,0}$ is a partitioned map.

Consider the terminology and notation of Example 10.26. Let $\mathcal{A}$ be a Set-category $\mathcal{A}$, and $\mathcal{A}^e$ be its extended category over Set. Each class $[M, P, f] \in \text{Deck}(\mathcal{A}^e)\backslash \mathcal{A}_0^e$ defines a canonical map $\iota_{[M,P,f]} : P = \bar{\pi}([M, P, f]) \to \text{Iso}(A_*)\backslash A_*$ by setting $\iota_{[M,P,f]}(p) = [M, x]$ for any $x \in f^{-1}(p)$; notice that this definition is independent of the representative $(M, P, f)$ and the point $x \in f^{-1}(p)$. Observe also that $\iota_{[M,P,f]}$ is injective if and only if $\text{Deck}(M, P, f)$ is the trivial group.

An $\mathcal{A}$-partitioned set is a triple $(X, \mathcal{R}, \sigma)$, briefly denoted by $X$, where $(X, \mathcal{R})$ is a partitioned set, and $\sigma$ is a map $X/\mathcal{R} \to \text{Deck}(\mathcal{A}^e)\backslash \mathcal{A}_0^e$ satisfying $\bar{\pi} \circ \sigma(C) = C$ for all $C \in X/\mathcal{R}$. A partitioned map between $\mathcal{A}$-partitioned sets is called an $\mathcal{A}$-partitioned map if its restrictions to the equivalence classes underly morphisms between representatives of the corresponding elements of $\text{Deck}(\mathcal{A}^e)\backslash \mathcal{A}_0^e$. A composition of $\mathcal{A}$-partitioned maps is an $\mathcal{A}$-partitioned map; thus $\mathcal{A}$-partitioned maps between small $\mathcal{A}$-partitioned sets form a Set-category.

Let $X \equiv (X, \mathcal{R}, \sigma)$ be an $\mathcal{A}$-partitioned set. A canonical map $\iota_X : X \to \text{Iso}(A_*)\backslash A_*$, briefly denoted by $\iota$, is defined by $\iota_X(x) = \iota_{\sigma(\mathcal{R}(x))}(x)$. The $\mathcal{A}$-partitioned set $X$ is said to be *aperiodic* when, for all $C_1, C_2 \in X/\mathcal{R}$, if there is an isomorphism $\psi$ between representatives of $\sigma(C_1)$ and $\sigma(C_2)$ in $\mathcal{A}^e$, then $C_1 = C_2$ and $\pi(\psi)$ is the identity map; this means that $\iota_X$ is injective.

The prototype example of aperiodic $\mathcal{A}$-partitioned set is the partitioned set $\text{Iso}(A_*)\backslash A_{*,0}$ endowed with the canonical map $\sigma : \text{Iso}(\mathcal{A})\backslash A_0 \to \text{Deck}(\mathcal{A}^e)\backslash \mathcal{A}_0^e$ defined by $\sigma([M]) = [M, \iota(M), \iota]$. In this example, the canonical map $\iota : \text{Iso}(A_*)\backslash A_{*,0} \to \text{Iso}(A_*)\backslash A_{*,0}$ is the identity map. Moreover $\iota_X$ is an $\mathcal{A}$-partitioned map for every $\mathcal{A}$-partitioned set $X$. Therefore $\text{Iso}(A_*)\backslash A_*$ is *universal* among all aperiodic $\mathcal{A}$-partitioned sets in the sense that there is an $\mathcal{A}$-partitioned injection of each of them into $\text{Iso}(A_*)\backslash A_*$.

**Example 10.123.** Any partitioned set $(X, \mathcal{R})$ with small equivalence classes defines a Set-partitioned set $(X, \mathcal{R}, \sigma)$ by setting $\sigma(C) = [C, C, \text{id}_C]$ for each $C \in X/\mathcal{R}$.

A *partitioned space* is a topological space $X$ endowed with an equivalence relation $\mathcal{R}$. Continuous partitioned maps between small partitioned spaces form a Top-category. For instance, any foliated space is a partitioned space, and any foliated map is a partitioned map.

Let $\mathcal{A}$ be a Top-category and $\mathcal{A}^e$ its extended category over Top, which can be identified to its extended category over Set. An $\mathcal{A}$-partitioned space is a triple $(X, \mathcal{R}, \sigma)$, where $(X, \mathcal{R})$ is a partitioned space and $(|X|_{\text{Set}}, \mathcal{R}, \sigma)$ is an $\mathcal{A}$-partitioned set, such that, for each $C \in X/\mathcal{R}$, if $\sigma(C) = [M, P, f]$, then the inclusion map $|P|_{\text{Set}} = C \hookrightarrow X$ underlies a local embedding $P \to X$. A composition of continuous $\mathcal{A}$-partitioned maps between $\mathcal{A}$-partitioned spaces is a continuous $\mathcal{A}$-partitioned map. Thus continuous $\mathcal{A}$-partitioned maps between small $\mathcal{A}$-partitioned spaces form a Top-category. The isomorphisms, embeddings, etc. of this category
will be called *A-partitioned homeomorphisms*, *A-partitioned embeddings*, etc. If some $M \in \mathcal{A}_0$
is locally aperiodic, then $\iota_M$ is locally injective. For any *A-partitioned space* $X \equiv (X, \mathcal{R}, \sigma)$,
*local aperiodicity* can be defined in the obvious way, and means that $\iota_X$ is locally injective.

Now, let $\mathcal{A}/\mathcal{B}$ be a u-functor over Top/ParSet, and let $\mathfrak{b}$ be a bidistortion of $\text{Iso}(\mathcal{A}_*)/\mathcal{B}^{\text{in}}$.
It is said that $\mathfrak{b}$ is continuous on some $M \in \mathcal{A}_0$ if, for each $x \in M$ and every $\varepsilon > 0$, there is
some neighborhood $U$ of $x$ in $M$ such that, for all $y \in U$, there are some $f,g \in \mathcal{B}(|M|,|M|)$
so that $f(x) = y, g(y) = x$ and $\mathfrak{b}_{M,x;M,y}(f,g) < \varepsilon$; this means that $\iota_M$ is continuous with
respect to the topology induced by $d_{\mathfrak{b}}$ on $\text{Iso}(\mathcal{A}_*)\backslash\mathcal{A}_{*,0}$. It is said that $\mathfrak{b}$ is continuous if it is
continuous on all $M \in \mathcal{A}_0$.

With more generality, it is said that **b** is *continuous* on an **A**-partitioned space *X* ≡ (*X*, *R*, *σ*) when *ι*<sub>*X*</sub> is continuous. In this case, *X* is called a **b-partitioned space**. Continuous *A*-partitioned maps between **b-partitioned spaces** are called **b-partitioned maps**. If only small **b-partitioned spaces** are considered, **b-partitioned maps** form a category. Concepts like **b-partitioned homeomorphisms**, **b-partitioned embeddings**, etc. have the obvious meanings.

**Proposition 10.124.** If the objects of *A* are locally compact, Iso(*A*<sub>*</sub>)\* *A*<sub>*</sub> is Hausdorff and **b**
is continuous, then the *A*-partitioned set Iso(*A*<sub>*</sub>)\* *A*<sub>*</sub> becomes a **b**-partitioned space with the
topology induced by d<sub>**b**</sub>.

*Proof.* For each $M \in \mathcal{A}^0$, endow $\iota(M)$ with the topology so that $\iota: M \to \iota(M)$ is an identification. The inclusion map $\iota(M) \hookrightarrow \text{Iso}(\mathcal{A}_*) \backslash \mathcal{A}_*$ is continuous because $\mathfrak{b}$ is continuous. On the other hand, since $\iota(M)$ can be considered as the quotient space of $M$ by canonical action of $\text{Iso}_A(M)$, the map $\iota: M \to \iota(M)$ is open. Therefore $\iota(M)$ is locally compact because so is $M$. It follows that $\iota(M) \hookrightarrow \text{Iso}(\mathcal{A}_*) \backslash \mathcal{A}_*$ is a local embedding since $\text{Iso}(\mathcal{A}_*) \backslash \mathcal{A}_*$ is Hausdorff. $\square$

When the conditions of Proposition 10.124 are satisfied, for any compact aperiodic **b**-
partitioned space *X*, the canonical map *ι*<sub>*X*</sub> is a **b**-partitioned embedding of *X* into Iso(*A*<sub>*</sub>)\* *A*<sub>*</sub>;
in this sense, it could be said that Iso(*A*<sub>*</sub>)\* *A*<sub>*</sub> is *universal* for compact aperiodic **b**-partitioned
spaces. If *X* is only locally compact, then *ι*<sub>*X*</sub> is a **b**-partitioned local embedding

**Example 10.125.** With the notation of Example 10.118, the restrictions of $\mathfrak{b}_{\text{lsl}}^{*\ddagger}$ and $\mathfrak{b}_{\text{lsl}}^{\ddagger\ddagger}$ to $\text{Iso}(\text{PropMet}_*) / \text{ParSet}^{\text{in}}$ satisfy the hypothesis of Proposition 10.124.

**Definition 10.126.** It is said that **b** is *stable* if, for every *δ* > 0, all (M, x), (N, y) ∈ A<sub>*<sub>,0</sub></sub>, all compact subspaces P ⊂ M and Q ⊂ N, and all compact-open neighborhoods U and V of P ↪ M and Q ↪ N, respectively, there is some ε > 0 such that, if **b**<sub>M,x;N,y</sub>(f, g) < ε for some (f, g) ∈ B<sup>in</sup>(M, x; N, y), then:

• $K \subset \text{dom } g \circ f$ and $L \subset \text{dom } f \circ g$;

• the restrictions $g \circ f : K \to M$ and $f \circ g : L \to N$ belong to $\mathcal{U}$ and $\mathcal{V}$, respectively;

• the restrictions $f: P \to N$ and $g: Q \to M$ are embeddings; and

• $\mathfrak{b}_{M,x';N,y'}(f,g) < \delta$ if $x' \in K$ and $y' = f(x')$, or $y' \in L$ and $x' = g(y')$.
# Chapter 11

# Universal foliated spaces

***

## 11.1 A universal foliated space

Fix any $n \in \mathbb{N}$. For $v \in \mathbb{R}^n$ and $r > 0$, let $B(v, r)$ be the open ball of $\mathbb{R}^n$ with center $v$ and radius $r$. The following terminology is introduced for any Hausdorff manifold $M$ of dimension $n$ and any continuous map $\xi: M \to \mathbb{R}^n$. For each $x \in M$, a number $r > 0$ is called an *admissible radius* of $\xi$ at $x$ if there is an open neighborhood $P$ of $x$ such that $\xi$ restricts to a homeomorphism $P \to B(\xi(x), r)$. Notice that this $P$ is the connected component of $\xi^{-1}(B(\xi(x), r))$ that contains $x$. It is called the *plaque* of $\xi$ with center $x$ and radius $r$, and denoted by $P_\xi(x, r)$. The *injectivity radius* of $\xi$ at $x$, denoted by $\iota_\xi(x)$, is the lowest upper bound in $[0, \infty]$ of the admissible radii of $\xi$ at $x$; in particular, $\iota_\xi(x) = 0$ if and only if $\xi$ has no admissible radius at $x$. The proof of the following result is elementary.

**Lemma 11.1.** If $\iota_\xi(x) > 0$, then the set of all admissible radii of $\xi$ at $x$ is $(0, \iota_\xi(x)]$. Moreover

$$ \iota_{\xi}(y) \geq \iota_{\xi}(x) - \|\xi(y) - \xi(x)\| $$

for any $y \in P(x, \iota_{\xi}(x))$, and

$$ P(y, r) = P(x, \iota_{\xi}(x)) \cap \xi^{-1}(B(\xi(y), r)) $$

for

$$ 0 < r \leq \iota_{\xi}(x) - \|\xi(y) - \xi(x)\|. $$

For $0 < r \leq s$, consider the family of plaques

$$ \mathcal{P}_{\xi}^{r,s} = \{P_{\xi}(x, r) \mid x \in \xi^{-1}(0), \iota_{\xi}(x) > s\}, $$

which are disjoint from each other, and whose union is denoted by $U_{\xi}^{r,s}$. Observe that

$$ (11.1) \qquad r' \leq r \leq s \leq s' \implies U_{\xi}^{r',s'} \subset U_{\xi}^{r,s}, $$

$$ (11.2) \qquad U_{\xi}^{r,s} = \bigcup_{r' < r, s' > s} U_{\xi}^{r',s'}. $$
Let $N$ be another Hausdorff manifold of dimension $n$, and consider continuous maps $\zeta : N \to \mathbb{R}^n$ and $\phi : M \to N$.

**Lemma 11.2.** For $x \in M$, if $\zeta \circ \phi = \xi$, then $\iota_{\zeta}(\phi(x)) \ge \iota_{\xi}(x)$. If moreover $\iota_{\xi}(x) > 0$, then $\phi(P_{\xi}(x, r)) = P_{\zeta}(\phi(x), r)$ and the restriction $\phi : P_{\xi}(x, r) \to P_{\zeta}(\phi(x), r)$ is a homeomorphism for $0 < r \le \iota_{\xi}(x)$.

*Proof.* The first inequality is obvious if $\iota_{\xi}(x) = 0$. So assume that $\iota_{\xi}(x) > 0$ and take $r \in (0, \iota_{\xi}(x)]$. Since $\xi$ restricts to a homeomorphism $P_{\xi}(x, r) \to B(\xi(x), r)$ and $\zeta \circ \phi = \xi$, it follows that $\zeta$ restricts to a continuous bijection $\phi(P_{\xi}(x, r)) \to B(\xi(x), r)$, and $\phi$ restricts to a continuous injection $P_{\xi}(x, r) \to N$. Moreover $\phi(P_{\xi}(x, r))$ is open in $N$ and $\zeta : \phi(P_{\xi}(x, r)) \to B(\xi(x), r)$ is open by the domain invariance theorem [?]. $\square$

Let $\mathcal{A}$ be the category defined as follows. Its objects are the elements of $\mathfrak{A}$ of the form $\mathbf{M} = (M, \xi, u, d)$, where

* $M$ is a small, connected, Hausdorff and second countable manifold $M$ of dimension $n$;

* $\xi$ is a sequence of continuous maps $\xi_i : M \to \mathbb{R}^n$ such that

- $\{U_{\xi_i}^{1,2} | i \in \mathbb{N}\}$ covers $M$,

- $\{U_{\xi_i}^{2,2} | i \in \mathbb{N}\}$ is locally finite, and

- if there is some connected open subset $P \subset M$ satisfies $\xi_i(P) = B(0, s)$ for some $i \in \mathbb{N}$ and some $s > 2$, then $P$ contains some plaque in $\mathcal{P}_{\xi_i}^{1,2}$;

* $u$ is a double sequence of locally constant functions $u_{i,j} : U_{\xi_i}^{1,2} \to [0, 1]$; and

* $d$ is a continuous proper coarse metric on $M$ such that

- $d \ge 1$,

- $d(x, y) = 1$ if and only if there is some $i \in \mathbb{N}$ and some $P \in \mathcal{P}_{\xi_i}^{1,2}$ such that $x, y \in \bar{P}$, and,

- if $d(x, y) < 2$, then there is some $i \in \mathbb{N}$ and some $Q \in \mathcal{P}_{\xi_i}^{2,2}$ such that $x, y \in Q$.

It may happen that $U_{\xi_i}^{1,2} = \emptyset$ for some $i$; in this case, the maps $u_{i,j}$ are empty. In fact, this is the case for almost all $i$ if $M$ is compact. The plaques of the maps $\xi_i$ are called *plaques of M*.

For each $x \in \mathbf{M}$ and $r \ge 0$, the “closed coarse ball” of center $x$ and radius $r$,

$$D_{\mathbf{M}}(x, r) = \{y \in \mathbf{M} \mid d(x, y) \le r\},$$

is compact because $d$ is continuous and proper. Notice that $D_{\mathbf{M}}(x, r) = \emptyset$ if $r < 1$.

For objects $\mathbf{M}_k = (M_k, \xi_k, u_k, d_k)$ of $\mathcal{A}$, where $k \in \{0, 1\}$, $\xi_k = \{\xi_{k,i}\}$ and $u_k = \{u_{k,i,j}\}$, a morphism $\mathbf{M}_0 \to \mathbf{M}_1$ of $\mathcal{A}$ is a continuous map $\phi : M_0 \to M_1$ such that
• $\xi_{0,i} = \xi_{1,i} \circ \phi$ for all $i \in \mathbb{N}$, yielding $\phi(U_{\xi_{0,i}}^{1,2}) \subset U_{\xi_{1,i}}^{1,2}$ by Lemma 11.2;

• $\iota_{\xi_{0,i}}(x) = \iota_{\xi_{1,i}}(\phi(x))$ for all $i \in \mathbb{N}$ and $x \in M_0$;

• $u_{0,i,j} = u_{1,i,j} \circ \phi$ for all $i, j \in \mathbb{N}$; and

• $d_0(x, y) = d_1(\phi(x), \phi(y))$ for all $x, y \in M_0$.

The operation of $\mathcal{A}$ is defined by composition.

**Proposition 11.3.** *Any morphism of $\mathcal{A}$ is a covering map.*

*Proof.* Consider a morphism $\phi \in \mathcal{A}(\mathbf{M}_0, \mathbf{M}_1)$ as above. For any $u \in \overline{\phi(\mathbf{M}_0)}$, there is some $i \in \mathbb{N}$ and plaque in $\mathcal{P}_{\xi_{1,i}}^{1,2}$ containing $u$; this plaque is of the form $P_{\xi_{1,i}}(v, 1)$ for some $v \in \mathbf{M}_1$ with $\iota_{\xi_{1,i}}(v) > 2$. Since $P_{\xi_{1,i}}(v, 1) \cap \phi(\mathbf{M}_0) \neq \emptyset$, there is some $x \in \mathbf{M}_0$ so that $\phi(x) \in P_{\xi_{1,i}}(v, 1)$; in particular, $\|\xi_{1,i}\phi(x) - \xi_{1,i}(u)\| < 1$. From Lemmas 11.2 and 11.1, it follows that $\iota_{\xi_{0,i}}(x) = \iota_{\xi_{1,i}}(\phi(x)) > 1$ and

$$ \phi(P_{\xi_{0,i}}(x, 1)) = P_{\xi_{1,i}}(\phi(x), 1) = P_{\xi_{1,i}}(v, 2) \cap \xi_{1,i}^{-1}(B(\xi_{1,i}\phi(x), 1)), $$

which contains $u$. Therefore $u \in \phi(\mathbf{M}_0)$, showing that $\phi(\mathbf{M}_0)$ is closed in $\mathbf{M}_1$.

Now, take any $x \in \mathbf{M}_0$. There is some $i \in \mathbb{N}$ and some plaque in $\mathcal{P}_{\xi_{0,i}}^{1,2}$ containing $x$; this plaque is of the form $P_{\xi_{0,i}}(y, 1)$ for some $y \in \mathbf{M}_0$ with $\iota_{\xi_{0,i}}(y) > 2$. By Lemma 11.2, we have $\iota_{\xi_{1,i}}(\phi(x)) = \iota_{\xi_{0,i}}(x) > 2$ and $P_{\xi_{1,i}}(\phi(x), 1) = \phi(P_{\xi_{0,i}}(x, 1))$. Hence $P_{\xi_{1,i}}(\phi(x), 1) \subset \phi(\mathbf{M}_0)$, showing that $\phi(\mathbf{M}_0)$ is also open in $\mathbf{M}_1$. Therefore $\phi$ is surjective because $\mathbf{M}_1$ is connected.

Finally, take any $i \in \mathbb{N}$ and any plaque in $\mathcal{P}_{\xi_{1,i}}^{1,2}$ of the form $P_{\xi_{1,i}}(u, 1)$. By Lemma 11.2, $\phi^{-1}(P_{\xi_{1,i}}(u, 1))$ is the disjoint union of the plaques $P_{\xi_{0,i}}(x, 1)$ with $x$ running in $\phi^{-1}(u)$. So $\phi$ is a covering map because $\bigcup_{i=1}^{\infty} \mathcal{P}_{\xi_{0,i}}^{1,2}$ covers $\mathbf{M}_0$. $\square$

**Proposition 11.4.** For $\mathbf{M} = (M, \xi, u, d) \in \mathcal{A}_0$, $\phi \in \mathcal{A}(\mathbf{M}, \mathbf{M})$, $i \in \mathbb{N}$ and $P \in \mathcal{P}_{\xi_i}^{1,2}$, if $\phi(P) \cap P \neq \emptyset$, then $\phi = \mathrm{id}_{\mathbf{M}}$; in particular, $\mathbf{M}$ is locally aperiodic.

*Proof.* By Lemma 11.2, $\phi(P) \in \mathcal{P}_{\xi_i}^{1,2}$. So $\phi$ is the identity on $P$ because the plaques in $\mathcal{P}_{\xi_i}^{1,2}$ are disjoint from each other. Hence $\phi = \mathrm{id}_{\mathbf{M}}$ since $\mathbf{M}$ is connected and covered by $\bigcup_{i=1}^{\infty} \mathcal{P}_{\xi_i}^{1,2}$. $\square$

**Corollary 11.5.** $\mathcal{A}_*(\mathbf{M}, x; \mathbf{M}, x) = \{\mathrm{id}_{\mathbf{M},x}\}$ for all $x \in \mathbf{M}$.

**Corollary 11.6.** If the functions $u_{i,j}$ separate the elements of $\mathcal{P}_{\xi_i}^{1,2}$ for each $i \in \mathbb{N}$, then $\mathcal{A}(\mathbf{M}, \mathbf{M}) = \{\mathrm{id}_{\mathbf{M}}\}$.

The category $\mathcal{A}$ is over $\mathrm{Man}(n)$ in a canonical way, and thus over $\mathcal{B} = \mathrm{LocMan}(n)$ too. Consider the induced $u$-functor $\mathcal{A}_*/\mathcal{B}$, and let $\tilde{\mathcal{B}}$ denote the corresponding lift of $\mathcal{B}$. Define a map $\mathfrak{d} : \tilde{\mathcal{B}} \to [0, \infty)$ in the following way. Let $(\mathbf{M}_k, x_k) \in \mathcal{A}_{*,0}$ for $k \in \{0, 1\}$ with $\mathbf{M}_k$ as above, and let $f \in \tilde{\mathcal{B}}(\mathbf{M}_0, x_0; \mathbf{M}_1, x_1)$. Let

$$ (11.3) \qquad \left\{ \begin{array}{ll} D_{k,r} = D_{\mathbf{M}_k}(x_k, r), & \iota_{k,i}(x) = \iota_{\xi_{k,i}}(x), \\ P_{k,i}(x, r) = P_{\xi_{k,i}}(x, r), & \mathcal{P}_{k,i}^{r,s} = \mathcal{P}_{\xi_{k,i}}^{r,s}, \quad U_{k,i}^{r,s} = U_{\xi_{k,i}}^{r,s}, \end{array} \right. $$
where $x \in M$ and $0 < r < s$. If $x_0 \notin \text{dom } f$, then set $\mathcal{D}(f) = \infty$. Otherwise, fixed monotone sequences of positive numbers, $r_q \uparrow 1$ and $s_q \downarrow 2$, and consider the following lowest upper bounds in $[0, \infty]$:

$$
\begin{align*}
R(f) &= \sup\{r \ge 0 \mid D_{0,r} \subset \text{dom } f\}, \\
\theta(f) &= \sup\{\|\xi_{1,i}(x_1) - \xi_{1,i}f(x_0)\| \mid i \in \mathbb{N}\}, \\
\alpha_p^1(f) &= \sup\{\|\xi_{0,i}(x) - \xi_{1,i}f(x)\| \mid i \in \mathbb{N}, x \in D_{0,p} \cap \text{dom } f\}, \\
\alpha_p^2(f) &= \sup\{|\iota_{0,i}(x) - \iota_{1,i}(f(x))| \mid i \in \mathbb{N}, x \in D_{0,p} \cap \text{dom } f\}, \\
\alpha_p^3(f) &= \sup\{|d_0(x, y) - d_1(f(x), f(y))| \mid x, y \in D_{0,p} \cap \text{dom } f\}, \\
\alpha_p^4(f) &= \sup\{|d_1(x_1, x) - d_1(f(x_0), x)| \mid x \in D_{0,p} \cap \text{dom } f\}, \\
\beta_{p,q}(f) &= \sup\{|u_{0,i,j}(x) - u_{1,i,j}f(x)| \mid i, j \in \mathbb{N}, x \in D_{0,p} \cap U_{0,i}^{r_q,s_q} \cap f^{-1}(U_{1,i}^{1,2})\},
\end{align*}
$$

where $p, q \in \mathbb{N}$. Then set

$$
\mathcal{D}(f) = \frac{1}{R(f)} + \theta(f) + \sum_{p=1}^{\infty} \frac{\min\{1, \alpha_p^1(f) + \dots + \alpha_p^4(f)\}}{2^p} + \sum_{p,q=1}^{\infty} \frac{\min\{1, \beta_{p,q}(f)\}}{2^{p+q}}.
$$

**Proposition 11.7.** *d* is a distortion of *A*\*/B.

*Proof.* It is obvious that $\mathcal{D}$ satisfies (d1).

With the above notation, if $\mathcal{D}(f) = 0$, then we get the following:

• $R(f) = \infty$, yielding $\text{dom } f = M_0$.

• $\alpha_p^1(f) = 0$ for all $p \in \mathbb{N}$, yielding $\xi_{0,i} = \xi_{1,i}f$ for all $i \in \mathbb{N}$, and thus $f(U_{0,i}^{1,2}) \subset U_{1,i}^{1,2}$ by Lemma 11.2.

• $\alpha_p^2(f) = 0$ for all $p \in \mathbb{N}$, yielding $\iota_{0,i}(x) = \iota_{1,i}(f(x))$ for all $i \in \mathbb{N}$ and all $x \in M_0$.

• $\alpha_p^3(f) = 0$ for all $p \in \mathbb{N}$, yielding $d_0(x, y) = d_1(f(x), f(y))$ for all $x, y \in M_0$.

• $\alpha_p^4(f) = 0$ for all $p \in \mathbb{N}$, yielding $d_1(x_1, f(x_0)) = d_1(f(x_0), f(x_0)) = 1$, which means that there is some $i_0 \in \mathbb{N}$ and some $P \in \mathcal{P}_{1,i_0}^{1,2}$ such that $x_1, f(x_0) \in \bar{P}$. Thus $x_1, f(x_0) \in Q$ for some $Q \in \mathcal{P}_{1,i_0}^{2,2}$.

• $\theta(f) = 0$, yielding $\xi_{1,i}(x_1) = \xi_{1,i}(f(x_0))$ for all $i \in \mathbb{N}$. So $f(x_0) = x_1$ since $\xi_{1,i_0} : Q \to \mathbb{R}^n$ is an embedding.

• $\beta_{p,q}(f) = 0$ for all $p, q \in \mathbb{N}$, yielding $u_{0,i,j} = u_{1,i,j}f$ for all $i, j \in \mathbb{N}$ by (11.1) and (11.2).

Therefore $f \in \mathcal{A}_*(\mathbf{M}_0, x_0; \mathbf{M}_1, x_1)$, which shows that $\mathcal{D}$ satisfies (d2).

Finally, let us prove that $\mathcal{D}$ satisfies (d3). Consider a finite sequence of objects $(\mathbf{M}_k, x_k) \in \mathcal{A}_{*,0}$ for $k \in \{0, \dots, \ell\}$, where $\mathbf{M}_k = (M_k, \xi_k, u_k, d_k)$, $\xi_k = \{\xi_{k,i}\}$ and $u_k = \{u_{k,i,j}\}$, and take morphisms $f_k \in \tilde{\mathcal{B}}(\mathbf{M}_{k-1}, x_{k-1}; \mathbf{M}_k, x_k)$ for $k > 0$. To make inductive arguments easier, let
also $f_0 = \mathrm{id}_{M_{0,x_0} \in \mathcal{A}_*(M_0, x_0; M_0, x_0)}$. Consider the notation (11.21) with $k \in \{0, \dots, \ell\}$, and let

$$
\begin{align*}
\hat{f}_k &= f_k \circ \dots \circ f_0 \\
R_k &= R(f_k), \qquad \hat{R}_k = R(\hat{f}_k), \\
\theta_k &= \theta(f_k), \qquad \hat{\theta}_k = \theta(\hat{f}_k), \\
\alpha_{k,p}^a &= \alpha_p^a(f_k), \qquad \hat{\alpha}_{k,p}^a = \alpha_p^a(\hat{f}_k), \\
\beta_{k,p,q} &= \beta_{p,q}(f_k), \qquad \hat{\beta}_{k,p,q} = \beta_{p,q}(\hat{f}_k),
\end{align*}
$$

where $a \in \{1, \dots, 4\}$ and $p, q \in \mathbb{N}$. Given $\varepsilon > 0$, there is some $\delta > 0$ such that, for some $m \in \mathbb{N}$, we have

$$
(11.4) \qquad \frac{1}{m} + (1 + 5 \cdot 2^{m+1} + 2^{m+2})\delta < \frac{\varepsilon}{2},
$$

$$
(11.5) \qquad \sum_{p=m+1}^{\infty} \frac{1}{2^p} + \sum_{p,q=m+1}^{\infty} \frac{1}{2^{p+q}} < \frac{\epsilon}{2},
$$

$$
(11.6) \quad 2^{m+1} \delta < \min\{r_1, r_2 - r_1, \dots, r_{m+1} - r_m, s_1 - s_2, \dots, s_m - s_{m+1}\}.
$$

Assume that $\mathfrak{d}(f_1) + \dots + \mathfrak{d}(f_\ell) < \delta$; in particular,

$$
(11.7) \qquad \frac{1}{R_1}, \dots, \frac{1}{R_\ell} < \delta,
$$

$$
(11.8) \qquad \theta_1 + \dots + \theta_\ell < \delta,
$$

$$
(11.9) \qquad \sum_{k=1}^{\ell} \sum_{p=1}^{m+1} (\alpha_{k,p}^{1} + \dots + \alpha_{k,m}^{4}) < 2^{m+1}\delta,
$$

$$
(11.10) \qquad \sum_{k=1}^{\ell} \sum_{p,q=1}^{m+1} \beta_{k,p,q} < 2^{2m+2}\delta.
$$

**Claim 33.** $\hat{f}_k(D_{0,p} \cap \mathrm{dom} \hat{f}_k) \subset D_{k,p+1}$ for $k \in \{0, \dots, \ell\}$ and $p \in \{1, \dots, m\}$.

Indeed, let us show that

$$
(11.11) \qquad \hat{f}_k(D_{0,p} \cap \mathrm{dom} \hat{f}_k) \subset D_{k,t_{k,p}},
$$

where

$$
t_{k,p} = p + \alpha_{0,p+1}^{3} + \alpha_{0,p+1}^{4} + \dots + \alpha_{k,p+1}^{3} + \alpha_{k,p+1}^{4},
$$

which is $< p + 1$ by (11.6) and (11.9). We proceed by induction on $k$. The inclusion (11.11)
is obvious for $k = 0$. Then take any $k > 0$ and suppose that (11.11) holds for $k - 1$. For any
$x \in D_{0,p}$, we have

$$
\begin{align*}
d_k(x_k, \hat{f}_k(x)) &\le d_{k-1}(x_{k-1}, \hat{f}_{k-1}(x)) \\
&\quad + |d_k(f_k(x_{k-1}), \hat{f}_k(x)) - d_{k-1}(x_{k-1}, \hat{f}_{k-1}(x))| \\
&\quad + |d_k(x_k, \hat{f}_k(x)) - d_k(f_k(x_{k-1}), \hat{f}_k(x))| \\
&\le t_{k-1,p} + \alpha_{k,p+1}^3 + \alpha_{k,p+1}^4,
\end{align*}
$$

yielding (11.11).

**Claim 34.** $\hat{R}_k \ge m$ for all $k \in \{0, \dots, \ell\}$.

Since $R_k > m + 1$ by (11.6) and (11.7), this assertion follows easily from Claim 33 by
induction on $k$.

**Claim 35.** We have

$$
\begin{align*}
\hat{\theta}_k &\le \theta_0 + \alpha_{0,1}^1 + \alpha_{0,2}^1 + \dots + \theta_k + \alpha_{k,1}^1 + \alpha_{k,2}^1, \\
\hat{\alpha}_{k,p}^1 &\le \alpha_{0,p+1}^1 + \dots + \alpha_{k,p+1}^1, \\
\hat{\alpha}_{k,p}^2 &\le \alpha_{0,p+1}^2 + \dots + \alpha_{k,p+1}^2, \\
\hat{\alpha}_{k,p}^3 &\le \alpha_{0,p+1}^3 + \dots + \alpha_{k,p+1}^3, \\
\hat{\alpha}_{k,p}^4 &\le 2\alpha_{0,p+1}^3 + \alpha_{0,p+1}^4 + \dots + 2\alpha_{k,p+1}^3 + \alpha_{k,p+1}^4,
\end{align*}
$$

for $k \in \{0, \dots, \ell\}$ and $p \in \{1, \dots, m\}$.

If $k > 0$, for any $i \in \mathbb{N}$ and $x, y \in D_{0,p} \cap \text{dom } \hat{f}_k$, we have $\hat{f}_{k-1}(x_0) \in D_{k-1,2}$ and
$\hat{f}_{k-1}(x), \hat{f}_{k-1}(y) \in D_{k-1,p+1}$ by Claim 33. Therefore

$$
\begin{align*}
\|\xi_{k,i}(x_k) - \xi_{k,i}\hat{f}_k(x_0)\| &\le \|\xi_{k,i}(x_k) - \xi_{k,i}\hat{f}_k(x_{k-1})\| \\
&\quad + \|\xi_{k,i}\hat{f}_k(x_{k-1}) - \xi_{k-1,i}(x_{k-1})\| \\
&\quad + \|\xi_{k-1,i}(x_{k-1}) - \xi_{k-1,i}\hat{f}_{k-1}(x_0)\| \\
&\quad + \|\xi_{k-1,i}\hat{f}_{k-1}(x_0) - \xi_{k,i}\hat{f}_k(x_0)\| \\
&\le \theta_k + \alpha_{k,1}^1 + \hat{\theta}_{k-1} + \alpha_{k,2}^1, \\[1em]
\|\xi_{0,i}(x) - \xi_{k,i}\hat{f}_k(x)\| &\le \|\xi_{0,i}(x) - \xi_{k-1,i}\hat{f}_{k-1}(x)\| \\
&\quad + \|\xi_{k-1,i}\hat{f}_{k-1}(x) - \xi_{k,i}\hat{f}_k(x)\| \\
&\le \hat{\alpha}_{k-1,p}^1 + \alpha_{k,p+1}^1, \\[1em]
|\iota_{0,i}(x) - \iota_{k,i}(\hat{f}_k(x))| &\le |\iota_{0,i}(x) - \iota_{k-1,i}(\hat{f}_{k-1}(x))| \\
&\quad + |\iota_{k-1,i}(\hat{f}_{k-1}(x)) - \iota_{k,i}(\hat{f}_k(x))| \\
&\le \hat{\alpha}_{k-1,p}^2 + \alpha_{k,p+1}^2, \\[1em]
|d_0(x, y) - d_k(\hat{f}_k(x), \hat{f}_k(y))| &\le |d_0(x, y) - d_{k-1}(\hat{f}_{k-1}(x), \hat{f}_{k-1}(y))| \\
&\quad + |d_{k-1}(\hat{f}_{k-1}(x), \hat{f}_{k-1}(y)) - d_k(\hat{f}_k(x), \hat{f}_k(y))| \\
&\le \hat{\alpha}_{k-1,p}^3 + \alpha_{k,p+1}^3, \\[1em]
|d_k(x_k, \hat{f}_k(x)) - d_k(\hat{f}_k(x_0), \hat{f}_k(x))| &\le |d_k(x_k, \hat{f}_k(x)) - d_k(f_k(x_{k-1}), \hat{f}_k(x))| \\
&\quad + |d_k(f_k(x_{k-1}), \hat{f}_k(x)) - d_{k-1}(x_{k-1}, \hat{f}_{k-1}(x))| \\
&\quad + |d_{k-1}(x_{k-1}, \hat{f}_{k-1}(x)) - d_{k-1}(\hat{f}_{k-1}(x_0), \hat{f}_{k-1}(x))| \\
&\quad + |d_{k-1}(\hat{f}_{k-1}(x_0), \hat{f}_{k-1}(x)) - d_k(\hat{f}_k(x_0), \hat{f}_k(x))| \\
&\le \alpha_{k,p+1}^4 + 2\alpha_{k,p+1}^3 + \hat{\alpha}_{k-1,p}^4,
\end{align*}
$$

yielding Claim 35 by induction on $k$.

**Claim 36.** $\hat{f}_k(D_{0,p} \cap U_{0,i}^{r,q,sq}) \subset U_{0,i}^{r,q+1,sq+1}$ for $k \in \{0, \dots, l\}$, $i \in \mathbb{N}$ and $p, q \in \{1, \dots, m\}$.

To prove this assertion, take any $x \in D_{0,p} \cap U_{0,i}^{r,q,sq}$. There is some $y \in \xi_{0,i}^{-1}(0)$ such that $\iota_{0,i}(y) > s_q$ and $x \in P_{0,i}(y, r_q)$. Thus $d_0(x, y) = 1$, yielding $y \in D_{0,p+1}$. Then

$$
\begin{align*}
\|\xi_{k,i}\hat{f}_k(y)\| &= \|\xi_{k,i}\hat{f}_k(y) - \xi_{0,i}(y)\| \le \hat{\alpha}_{k,p+1}^1, \\
\iota_{k,i}(\hat{f}_k(y)) &\ge \iota_{0,i}(y) - |\iota_{k,i}(\hat{f}_k(y)) - \iota_{0,i}(y)| > s_q - \hat{\alpha}_{k,p+1}^2.
\end{align*}
$$

Then

$$
B(0, s_q - \hat{\alpha}_{k,p+1}^2 - \hat{\alpha}_{k,p+1}^1) \subset B(\xi_{k,i}\hat{f}_k(y), s_q - \hat{\alpha}_{k,p+1}^2),
$$

and thus there exists a unique $y' \in \xi_{k,i}^{-1}(0)$ so that

$$
\iota_{k,i}(y') > s_q - \hat{\alpha}_{k,p+1}^2 - \hat{\alpha}_{k,p+1}^1, \quad \hat{f}_k(y) \in \overline{P_{k,i}(y', \hat{\alpha}_{k,p+1}^1)}.
$$
So

$$ (11.12) \qquad \iota_{k,i}(y') > s_{q+1}, $$

$$ (11.13) \qquad \hat{f}_k(y) \in P_{k,i}(y', r_{q+1}) $$

since

$$ \hat{\alpha}_{k,p+1}^2 + \hat{\alpha}_{k,p+1}^1 < s_q - s_{q+1}, \quad \hat{\alpha}_{k,p+1}^1 < r_1 $$

by Claim 35, (11.6) and (11.9). From (11.12), it follows that

$$ (11.14) \qquad P_{k,i}(y', r_{q+1}) \subset U_{k,i}^{r_{q+1}, s_{q+1}}. $$

Suppose that $\hat{f}_k(x) \notin U_{k,i}^{r_{q+1}, s_{q+1}}$. Hence $\hat{f}_k(x) \notin P_{k,i}(y', r_{q+1})$ by (11.14). Then, by (11.13), and since $P_{0,i}(y, r_q)$ is connected and contains $x$ and $y$, there is some $z \in P_{0,i}(y, r_q)$ such that $\hat{f}_k(z) \in \partial P_{k,i}(y', r_{q+1})$, yielding $\|\xi_{k,i}\hat{f}_k(z)\| = r_{q+1}$. Therefore

$$ \|\xi_{k,i}\hat{f}_k(z) - \xi_{0,i}(z)\| \ge \|\xi_{k,i}\hat{f}_k(z)\| - \|\xi_{0,i}(z)\| > r_{q+1} - r_q > \hat{\alpha}_{k,p+1}^1 $$

by Claim 35, (11.6) and (11.9), which is a contradiction. So $\hat{f}_k(x) \in U_{k,i}^{r_{q+1}, s_{q+1}}$, completing the proof of Claim 36.

**Claim 37.** We have

$$ \hat{\beta}_{k,p,q} \le \beta_{0,p+1,q+1} + \dots + \beta_{k,p+1,q+1} $$

$$ \text{for } k \in \{0, \dots, \ell\} \text{ and } p \in \{1, \dots, m\}. $$

If $k > 0$, for any $x \in D_{0,p} \cap U_{0,i}^{r_{q,sq}} \cap \hat{f}_k^{-1}(U_{k,i}^{1,2})$, we get $\hat{f}_{k-1}(x) \in D_{k-1,p+1} \cap U_{k-1,i}^{r_{q+1}, s_{q+1}}$ by Claims 33 and 36. Hence

$$ \begin{aligned} |u_{0,i,j}(x) - u_{k,i,j}\hat{f}_k(x)| &\le |u_{0,i,j}(x) - u_{k-1,i,j}\hat{f}_{k-1}(x)| \\ &\quad + |u_{k-1,i,j}\hat{f}_{k-1}(x) - u_{k,i,j}\hat{f}_k(x)| \\ &\le \hat{\beta}_{k-1,p,q} + \beta_{k,p+1,q+1}. \end{aligned} $$

Then Claim 37 follows by induction on $k$.
Now, from Claims 45, 35 and 37, and by (11.4), (11.5), (11.8), (11.9) and (11.10), we get

$$
\begin{align*}
\mathfrak{d}(\hat{f}_{\ell}) &= \frac{1}{\hat{R}_{\ell}} + \hat{\theta}_{\ell} + \sum_{p=1}^{\infty} \frac{\min\{1, \hat{\alpha}_{\ell,p}^{1} + \dots + \hat{\alpha}_{\ell,p}^{4}\}}{2^p} + \sum_{p,q=1}^{\infty} \frac{\min\{1, \hat{\beta}_{\ell,p,q}\}}{2^{p+q}} \\
&\le \frac{1}{m} + \sum_{k=0}^{\ell} (\theta_k + \alpha_{k,1}^1 + \alpha_{k,2}^1) \\
&\quad + \sum_{p=1}^{m} \sum_{k=0}^{\ell} \frac{\alpha_{k,p+1}^1 + \alpha_{k,p+1}^2 + 3\alpha_{k,p+1}^3 + \alpha_{k,p+1}^4}{2^p} \\
&\quad + \sum_{p=m+1}^{\infty} \frac{1}{2^p} + \sum_{p,q=1}^{m} \sum_{k=0}^{\ell} \frac{\beta_{k,p+1,q+1}}{2^{p+q}} + \sum_{p,q=m+1}^{\infty} \frac{1}{2^{p+q}} \\
&< \frac{1}{m} + \delta + 5 \cdot 2^{m+1}\delta + 2^{2m+2}\delta + \frac{\varepsilon}{2} \\
&< \varepsilon. \quad \Box
\end{align*}
$$

According to Example 10.107, Proposition 11.16 and Corollary 11.5, we get a good bidistortion **b** of Iso($\mathcal{A}_*$)/$\mathcal{B}^{\text{in}}$ defined by

$$
\mathfrak{b}(f,g) = \|(\mathfrak{d}(f), \mathfrak{d}(g))\|^*,
$$

where $\| \cdot \|$ is any fixed norm on $\mathbb{R}^2$ invariant by permutation of components. Nevertheless, $\text{Iso}(\mathcal{A}_*)/\mathcal{B}^{\text{in}}$ is not complete with $d_*$, and thus **b** is not complete by Proposition 10.92. The reason is that a uniform limit of homeomorphisms to a ball of $\mathbb{R}^n$ may not be a homeomorphism. Then, to show that $\text{Iso}(\mathcal{A}_*)/\mathcal{B}^{\text{in}}$ is Polish, let us embed it first in a larger space constructed in a similar way: instead of using homeomorphisms to balls of $\mathbb{R}^n$ to define plaques, we will use a larger class of maps which is preserved by uniform limits.

Let $X$ and $Y$ be topological spaces. A continuous, surjective and proper map $\phi : X \to Y$ is called a *pseudo-homeomorphism* when $\phi^{-1}(V)$ is connected for all connected open subset $V \subset N$. This last condition is kind of a version of injectivity. It is evident that any composition of two pseudo-homeomorphisms is a pseudo-homeomorphism.

**Example 11.8.** A continuous surjective map $\mathbb{R} \to \mathbb{R}$ is a pseudo-homeomorphism if and only if it is non-strictly monotone. This condition is slightly weaker than being strictly monotone, which means the injectivity.

The following result is obvious.

**Proposition 11.9.** If $f: X \to Y$ is a pseudo-homeomorphism, then the restriction $f: f^{-1}(W) \to W$ is a pseudo-homeomorphism for any open subset $W \subset Y$.

**Proposition 11.10.** Suppose that $Y$ is locally connected, and let $\beta$ be a base of open subsets of $Y$ which are connected. A continuous, surjective and proper map $f: X \to Y$ is a pseudo-homeomorphism if and only if $f^{-1}(B)$ is connected for all $B \in \beta$.
*Proof.* It is enough to prove the “if” part. Suppose that $f$ is not a pseudo-homeomorphism. Then there is some connected open subset $V \subset Y$ such that $f^{-1}(V)$ is not connected. Let $\{U_1, U_2\}$ be a non-trivial separation of $f^{-1}(V)$. If the sets $\text{Cl}_V(f(U_1))$ and $\text{Cl}_V(f(U_2))$ were disjoint, they would form a non-trivial separation of the connected set $V$. Therefore there is some point $y$ in the intersection of $\text{Cl}_V(f(U_1))$ and $\text{Cl}_V(f(U_2))$. Take some $B \in \beta$ so that $y \in B \subset V$. Then $B \cap f(U_1)$ and $B \cap f(U_2)$ are non-empty. So $f^{-1}(B) \cap U_1$ and $f^{-1}(B) \cap U_2$ are non-empty. Thus these sets form a non-trivial separation of $f^{-1}(B)$, which contradicts its connectivity. $\square$

**Lemma 11.11.** Let $f : X \to Y$ be a continuous map. Let $U$ and $V$ be open subsets of $X$ and $Y$, respectively, such that $f(U) = V$ and the restriction $f : U \to V$ is proper. If $X$ is Hausdorff, $Y$ is locally compact and $U$ is connected, then $U$ is a connected component of $f^{-1}(V)$.

*Proof.* Suppose that $U$ is not a connected component of $f^{-1}(V)$. Then there is some point $x$ in the boundary of $U$ so that $f(x) \in V$. Since $Y$ is locally compact, $f(x)$ has some compact neighborhood $Q$ contained in $V$. Hence $U \cap f^{-1}(Q)$ is compact and non-empty. Moreover $x \in \bar{U} \cap f^{-1}(Q)$ because $x \in \bar{U}$ and $f^{-1}(Q)$ is a neighborhood of $x$. Therefore $x \in U \cap f^{-1}(Q)$ because $X$ is Hausdorff. But $x \notin U$ because $U$ is open and $x$ is in its boundary, obtaining a contradiction. $\square$

**Lemma 11.12.** Let $f : X \to Y$ be a continuous map, and let $W_1 \subset W_2 \subset \dots$ be an exhausting increasing sequence of open subsets of $Y$. Let $c \in X$ such that $f(c) \in W_1$, and let $C$ be a connected component of $f^{-1}(Y)$ containing $c$. For each $k \in \mathbb{N}$, let $A_k$ be the connected component of $C \cap f^{-1}(W_k)$ containing $c$, and let $A = \bigcup_{k=1}^{\infty} A_k$. If $X$ is locally connected, then $A = C$.

*Proof.* The set $A$ is connected, contains $c$ and is contained in $f^{-1}(Y)$. Thus $A \subset C$. For each $k \in \mathbb{N}$, let $B_k$ be the union of connected components of $C \cap f^{-1}(W_k)$ which do not meet $A$, and let $B = \bigcup_{k=1}^{\infty} B_k$. It is easy to check that $A \cup B = C$ and $A \cap B = \emptyset$. Moreover $A$ and $B$ are open because $X$ is locally connected. Therefore $A = C$ because $C$ is connected and $c \in A$. $\square$

**Proposition 11.13.** Let $f_i$ be a sequence of continuous maps $X \to \mathbb{R}^n$ converging uniformly to a map $f$. Let $W$ be a connected open subset of $\mathbb{R}^n$, and let $C_i$ be a sequence of connected open subsets of $X$ contained in a compact subset $K$. Fix points $w \in W$ and $c_i \in C_i$ such that $f_i(c_i) = w$. Suppose that $X$ is Hausdorff and locally connected, the sequence $c_i$ converges to a point $c$, and the restrictions $f_i : C_i \to W$ are pseudo-homeomorphisms. Let $C$ be connected component of $f^{-1}(W)$ containing $c$. Then $C \subset K$ and the restriction $f : C \to W$ is a pseudo-homeomorphism.

*Proof.* The map $f$ is continuous because it is a uniform limit of continuous maps.

There is a sequence of connected open subsets $W_k \subset W$ such that $w \in W_k$, $\overline{W_k}$ is compact, $\overline{W_k} \subset W_{k+1}$ and $W = \bigcup_{k=1}^{\infty} W_k$. Then, for each $k \in \mathbb{N}$, there is some $\varepsilon_k > 0$ such
that $d(W_k, \mathbb{R}^n \setminus W_{k+1}) > \varepsilon_k$, where $d$ denotes the Euclidean distance. Moreover there is some $i'_k \in \mathbb{N}$ so that $d(\xi(x), \xi_i(x)) < \varepsilon_k$ for all $i \ge i'_k$ and all $x \in X$. It easily follows that

$$ (11.15) \qquad f^{-1}(W_k) \subset f_i^{-1}(W_{k+1}), \qquad f_i^{-1}(W_k) \subset f^{-1}(W_{k+1}). $$

Let $A_k$ be the connected component of $f^{-1}(W_k)$ that contains $c$. Observe that $A_k \subset C$. Since $A_k$ is open because $X$ is locally connected, there is some $i''_k \in \mathbb{N}$ such that $c_i \in A_k$ for all $i \ge i''_k$. By Lemma 11.11, each $C_i$ is the connected component of $f_i^{-1}(W)$ that contains $c_i$. So each $C_i \cap f_i^{-1}(W_k)$ is the connected component of $f_i^{-1}(W_k)$ that contains $c_i$ because $f_i: C_i \to W$ is a pseudo-homeomorphism. Hence

$$ (11.16) \qquad A_k \subset C_i \cap f_i^{-1}(W_{k+1}), \qquad C_i \cap f_i^{-1}(W_k) \subset A_{k+1} $$

for $i \ge i_k = \max\{i'_k, i''_k\}$ by (11.15).

**Claim 38.** $W_k \subset f(A_{k+1})$ for each $k \in \mathbb{N}$.

Take any $y \in W_k$. Since $f_i(C_i) = W$ for each $i \in \mathbb{N}$, there is some $x_i \in C_i \cap f_i^{-1}(W_k)$ such that $f_i(x_i) = y$. We have $x_i \in A_{k+1}$ for $i \ge i_k$ by (11.16), and we have

$$ d(f(x_i), y) = d(f(x_i), f_i(x_i)) \to 0 $$

as $i \to \infty$. This shows that $W_k \subset \overline{f(A_{k+1})}$. But $\overline{A_{k+1}}$ is compact because $X$ is Hausdorff and $A_{k+1} \subset K$ by (11.16). Hence $W_k \subset f(\overline{A_{k+1}})$, yielding

$$ W_k \subset f(f^{-1}(W_k) \cap \overline{A_{k+1}}) \subset f(f^{-1}(W_{k+1}) \cap \overline{A_{k+1}}). $$

On the other hand,

$$ f^{-1}(W_{k+1}) \cap \overline{A_{k+1}} = A_{k+1} $$

because $A_{k+1}$ is the connected component of $f^{-1}(W_{k+1})$ that contains $c$, and Claim 38 follows.

Lemma 11.12 and Claim 38 yield $f(C) = W$. Moreover

$$ C = \bigcup_{k=1}^{\infty} A_k \subset \bigcup_{k=1}^{\infty} C_{i_k} \subset K $$

by Lemma 11.12 and (11.16).

Let $Q$ be a compact subset of $W$. The set $f^{-1}(Q)$ is closed in $X$, and thus $C \cap f^{-1}(Q)$ is also closed in $X$ because $C$ is a connected component of $f^{-1}(W)$. Therefore $C \cap f^{-1}(Q)$ is compact since $C \subset K$. This shows that $f: C \to W$ is proper.

It remains to prove that $C \cap f^{-1}(V)$ is connected for any connected open subset $V \subset W$. By Proposition 11.10, we can assume $V \subset W_k$ for some $k \in \mathbb{N}$.

**Claim 39.** $A_\ell \cap f^{-1}(V)$ is connected for $\ell \ge k+1$.
Suppose that $A_\ell \cap f^{-1}(V)$ is not connected, and let $\{U_1, U_2\}$ be a non-trivial separation of this set. We have

$$f(U_1) \cup f(U_2) = f(A_\ell \cap f^{-1}(V)) = V$$

by Claim 38. So

$$\mathrm{Cl}_V(f(U_1)) \cap \mathrm{Cl}_V(f(U_2)) \neq \emptyset$$

because $V$ is connected. Take a point $y$ in this intersection, take some $\varepsilon > 0$ such that $B(y, 3\varepsilon) \subset V$, and take some integer $i \ge i_\ell$ such that $d(f(x), f_i(x)) < \varepsilon$ for all $x \in M$. For $a \in \{1, 2\}$, since $y \in \mathrm{Cl}_V(f(U_a))$, there is some $x \in U_a$ such that $d(y, f(x)) < \varepsilon$. We have

$$d(y, f_i(x)) \le d(y, f(x)) + d(f(x), f_i(x)) < 2\varepsilon,$$

yielding

$$(11.17) \qquad U_a \cap f_i^{-1}(B(y, 2\varepsilon)) \neq \emptyset.$$

On the other hand, for $x \in f_i^{-1}(B(y, 2\varepsilon))$, we get

$$d(y, f(x)) \le d(y, f_i(x)) + d(f_i(x), f(x)) < 3\varepsilon,$$

yielding

$$f(x) \in B(y, 3\varepsilon) \subset V.$$

So

$$f_i^{-1}(B(y, 2\varepsilon)) \subset f^{-1}(V),$$

yielding

$$C_i \cap f_i^{-1}(B(y, 2\varepsilon)) \subset A_\ell \cap f^{-1}(V) = U_1 \cup U_2$$

by (11.16). Therefore the sets $U_a \cap f_i^{-1}(B(y, 2\varepsilon))$, $a \in \{1, 2\}$, form a separation of $C_i \cap f_i^{-1}(B(y, 2\varepsilon))$, which is non-trivial by (11.17). This is a contradiction because $f_i : C_i \to W$ is a pseudo-homeomorphism, completing the proof of Claim 39.

Now,

$$C \cap f^{-1}(V) = \bigcup_{\ell=k+1}^{\infty} (A_\ell \cap f^{-1}(V))$$

by Lemma 11.12, and this union is connected by Claim 39. $\square$

<figure><img src="image_32.png" /></figure>

Let $M$ be a Hausdorff manifold of dimension $n$, and let $\xi : M \to \mathbb{R}^n$ be a continuous map. For each $x \in M$, a number $r > 0$ is called a *pseudo-admissible radius* of $\xi$ at $x$ if $\xi : P' \to B(\xi(x), r)$ is a pseudo-homeomorphism, where $P'$ is the connected component of $\xi^{-1}(B(\xi(x), r))$ that contains $x$. This $P'$ is called the *pseudo-plaque* of $\xi$ with center $x$ and radius $r$, and denoted by $P'_\xi(x, r)$. Observe that the center of a pseudo-plaque may not be unique. The *pseudo-injectivity radius* of $\xi$ at $x$, denoted by $\iota'_\xi(x)$, is the lowest upper bound in $[0, \infty]$ of the pseudo-admissible radii of $\xi$ at $x$. Notice that $\iota'_\xi(x) \ge \iota_\xi(x)$.
**Lemma 11.14.** If $\iota'_{\xi}(x)) > 0$, then the set of all pseudo-admissible radii of $\xi$ at $x$ is $(0, \iota'_{\xi}(x)]$. Moreover

$$ \iota'_{\xi}(y) \ge \iota'_{\xi}(x) - \|\xi(y) - \xi(x)\| $$ 
for any $y \in P'_{\xi}(x, \iota'_{\xi}(x))$, and

$$ P'_{\xi}(y, r) = P'_{\xi}(x, \iota'_{\xi}(x)) \cap \xi^{-1}(B(\xi(y), r)) $$ 
for

$$ 0 < r \le \iota'_{\xi}(x) - \|\xi(y) - \xi(x)\|. $$

*Proof.* There is a sequence $r_i \uparrow \iota'_{\xi}(x)$ consisting of pseudo-admissible radius of $\xi$ at $x$. Let $P' = \bigcup_{i=1}^{\infty} P'_{\xi}(x, r_i)$. We have $\xi(P') = B(\xi(x), \iota'_{\xi}(x))$ because $\xi(P'_{\xi}(x, r_i)) = B(\xi(x), r_i)$ for all $i \in \mathbb{N}$.

**Claim 40.** $P' \cap \xi^{-1}(V)$ is connected for any connected open subset $V$ of each $B(\xi(x), r_i)$.

This assertion holds because

$$ P' \cap \xi^{-1}(V) = \bigcup_{j=i}^{\infty} (P'(x, r_j) \cap \xi^{-1}(V)), $$

where each $P'(x, r_j) \cap \xi^{-1}(V)$ is connected since $\xi : P'(x, r_j) \to B(\xi(x), r_j)$ is a pseudo-homeomorphism.

**Claim 41.** For each $v \in B(\xi(x), \iota'_{\xi}(x))$, there is some $s > 0$ such that $\overline{B(v, s)} \subset B(\xi(x), \iota'_{\xi}(x))$ and $P' \cap \xi^{-1}(\overline{B(v, s)})$ is compact.

Take some $i \in \mathbb{N}$ such that $v \in B(\xi(x), r_i)$, and let $s_0 = r_i - \|v - \xi(x)\| > 0$. We have $\overline{B(v, s)} \subset B(\xi(x), r_i)$ for $0 < s < s_0$, and thus $P'_{\xi}(x, r_i) \cap \xi^{-1}(\overline{B(v, s)})$ is compact since $\xi : P'_{\xi}(x, r_i) \to B(\xi(x), r_i)$ is proper, yielding

$$ (11.18) \qquad \overline{P'_{\xi}(x, r_i) \cap \xi^{-1}(B(v, s))} \subset \overline{P'_{\xi}(x, r_i) \cap \xi^{-1}(\overline{B(v, s)})} \subset \overline{P'_{\xi}(x, r_i)}. $$

Suppose that $P' \cap \xi^{-1}(\overline{B(v, s)})$ is not compact for all $s \in (0, s_0)$. Then

$$ P' \cap \xi^{-1}(\overline{B(v, s)}) \subsetneq P'_{\xi}(x, r_i) $$

for all $s \in (0, s_0)$ because $P'_{\xi}(x, r_i) \cap \xi^{-1}(\overline{B(v, s)})$ is compact; indeed, we get

$$ P' \cap \xi^{-1}(\overline{B(v, s)}) \subsetneq \overline{P'_{\xi}(x, r_i)} $$

because $\overline{B(v, s/2)} \subset B(v, s)$. Moreover

$$ P'_{\xi}(x, r_i) \cap \xi^{-1}(\overline{B(v, s)}) \neq \emptyset $$
because $\xi(P'_{\xi}(x, r_i)) = B(\xi(x), r_i)$. Therefore

$$ \overline{P'_{\xi}(x, r_i) \cap \xi^{-1}(B(v, s))} \subset P'_{\xi}(x, r_i) $$

since $P' \cap \xi^{-1}(B(v, s))$ is connected by Claim 40. This contradicts (11.18), which proves Claim 41.

Since any compact subset of $B(\xi(x), l'_{\xi}(x))$ can be covered by a finite number of closed balls satisfying the conditions of Claim 41, it follows that $\xi: P' \to B(\xi(x), l'_{\xi}(x))$ is proper. Moreover this map is a pseudo-homeomorphism by Claim 40 and Proposition 11.10. Therefore $l'_{\xi}(x)$ is an admissible radius of $\xi$ at $x$.

Now, the rest of the statement follows easily from Proposition 11.9. $\square$

For $0 < r \le s$, consider the family of pseudo-plaques

$$ \mathcal{P}_{\xi}^{r,s} = \{P_{\xi}(x, r) \mid x \in \xi^{-1}(0), l'_{\xi}(x) > s\}, $$

which are disjoint from each other, and whose union is denoted by $U_{\xi}^{r,s}$. Observe that

$$ (11.19) \qquad r' \le r \le s \le s' \implies U_{\xi}^{r',s'} \subset U_{\xi}^{r,s}, $$

$$ (11.20) \qquad U_{\xi}^{r,s} = \bigcup_{r' < r, s' > s} U_{\xi}^{r',s'}. $$

Let $N$ be another Hausdorff manifold of dimension $n$, and consider continuous maps $\zeta: N \to \mathbb{R}^n$ and $\phi: M \to N$.

**Lemma 11.15.** For $x \in M$, if $\zeta \circ \phi = \xi$ and $l'_{\zeta}(\phi(x)) \ge l'_{\xi}(x) > 0$, then $\phi(P'_{\xi}(x, r)) \subset P'_{\zeta}(\phi(x), r)$ for $0 < r \le l'_{\zeta}(x)$.

*Proof.* By Lemma 11.11, the sets $P'_{\xi}(x, r)$ and $P'_{\zeta}(\phi(x), r)$ are the connected components of $\xi^{-1}(B(\xi(x), r))$ and $\zeta^{-1}(B(\zeta\phi(x), r))$ that contain $x$ and $\phi(x)$, respectively. Then the result follows because

$$ \phi(\xi^{-1}(B(\xi(x), r))) \subset \zeta^{-1}(B(\zeta\phi(x), r)) $$

since $\zeta \circ \phi = \xi$.

<figure><img src="image_33.png" /></figure>

Let $\mathcal{A}'$ be the category defined as follows. Its objects are the elements of $\mathfrak{A}$ of the form $M' = (M, \xi, u, d)$, where

* $M$ is a small, connected, Hausdorff and second countable manifold $M$ of dimension $n$;

* $\xi$ is a sequence of continuous maps $\xi_i: M \to \mathbb{R}^n$ such that $\{U_{\xi_i}^{1,2} \mid i \in \mathbb{N}\}$ covers $M$;

* $u$ is a double sequence of locally constant functions $u_{i,j}: U_{\xi_i}^{1,2} \to [0, 1]$; and

* $d$ is a continuous proper coarse metric on $M$ such that
$$-d \geq 1,$$

$$-d(x, y) = 1 \text{ if and only if there is some } i \in \mathbb{N} \text{ and some } P \in \mathcal{P}_{\xi_i}^{1,2} \text{ such that } x, y \in \bar{P}, \text{ and,}$$

$$- \text{ if } d(x, y) < 2, \text{ then there is some } i \in \mathbb{N} \text{ and some } Q \in \mathcal{P}_{\xi_i}^{1,2} \text{ such that } x, y \in Q.$$

The pseudo-plaques of the maps $\xi_i$ are called pseudo-plaques of $\mathbf{M}$.

For objects $\mathbf{M}'_k = (M_k, \xi_k, u_k, d_k)$ of $\mathcal{A}'$, where $k \in \{0, 1\}$, $\xi_k = \{\xi_{k,i}\}$ and $u_k = \{u_{k,i,j}\}$, a morphism $\mathbf{M}'_0 \to \mathbf{M}'_1$ of $\mathcal{A}'$ is a continuous map $\phi : M_0 \to M_1$ such that

$$\bullet \ \xi_{0,i} = \xi_{1,i} \circ \phi \text{ for all } i \in \mathbb{N};$$

$$\bullet \ \iota_{\xi_{0,i}}(x) = \iota_{\xi_{1,i}}(\phi(x)) \text{ for all } i \in \mathbb{N} \text{ and } x \in M_0, \text{ and thus } \phi(U_{\xi_{0,i}}^{1,2}) \subset U_{\xi_{1,i}}^{1,2} \text{ by Lemma 11.15};$$

$$\bullet \ u_{0,i,j} = u_{1,i,j} \circ \phi \text{ for all } i,j \in \mathbb{N}; \text{ and}$$

$$\bullet \ d_0(x, y) = d_1(\phi(x), \phi(y)) \text{ for all } x, y \in M_0.$$

The operation of $\mathcal{A}'$ is defined by composition.

The category $\mathcal{A}$ is over $\text{Man}(n)$ in a canonical way, and thus over $\mathcal{B} = \text{LocMan}(n)$ too. Consider the induced u-functor $\mathcal{A}_* / \mathcal{B}$, and let $\tilde{\mathcal{B}}$ denote the corresponding lift of $\mathcal{B}$. Define a map $\mathfrak{d} : \tilde{\mathcal{B}} \to [0, \infty]$ in the following way. Let $(\mathbf{M}_k, x_k) \in \mathcal{A}_{*,0}$ for $k \in \{0, 1\}$ with $\mathbf{M}_k$ as above, and let $f \in \tilde{\mathcal{B}}(\mathbf{M}_0, x_0; \mathbf{M}_1, x_1)$. Let

$$ (11.21) \qquad \left\{ \begin{array}{ll} D_{k,r} = D_{\mathbf{M}_k}(x_k, r), & \iota_{k,i}(x) = \iota_{\xi_{k,i}}(x), \\ P_{k,i}(x,r) = P_{\xi_{k,i}}(x,r), & \mathcal{P}_{k,i}^{r,s} = \mathcal{P}_{\xi_{k,i}}^{r,s}, \quad U_{k,i}^{r,s} = U_{\xi_{k,i}}^{r,s}, \end{array} \right. $$

where $x \in M$ and $0 < r < s$. If $x_0 \notin \text{dom } f$, then set $\mathfrak{d}(f) = \infty$. Otherwise, consider the following lowest upper bounds in $[0, \infty]$:

$$ \begin{align*} R(f) &= \sup\{r \ge 0 \mid D_{0,r} \subset \text{dom } f\}, \\ \theta(f) &= \sup\{\|\xi_{1,i}(x_1) - \xi_{1,i}f(x_0)\| \mid i \in \mathbb{N}\}, \\ \alpha_p^1(f) &= \sup\{\|\xi_{0,i}(x) - \xi_{1,i}f(x)\| \mid i \in \mathbb{N}, x \in D_{0,p} \cap \text{dom } f\}, \\ \alpha_p^2(f) &= \sup\{\lvert\iota_{0,i}(x) - \iota_{1,i}(f(x))\rvert \mid i \in \mathbb{N}, x \in D_{0,p} \cap \text{dom } f\}, \\ \alpha_p^3(f) &= \sup\{\lvert d_0(x, y) - d_1(f(x), f(y)) \rvert \mid x, y \in D_{0,p} \cap \text{dom } f\}, \\ \alpha_p^4(f) &= \sup\{\lvert d_1(x_1, x) - d_1(f(x_0), x) \rvert \mid x \in D_{0,p} \cap \text{dom } f\}, \\ \beta_{p,q}(f) &= \sup\{\lvert u_{0,i,j}(x) - u_{1,i,j}f(x) \rvert \mid i,j \in \mathbb{N}, x \in D_{0,p} \cap U_{0,i}^{r_q,s_q} \cap f^{-1}(U_{1,i}^{1,2})\}, \end{align*} $$

where $p, q \in \mathbb{N}$, and where we have fixed monotone sequences of positive numbers, $r_q \uparrow 1$ and $s_q \downarrow 2$. Then set

$$ \mathfrak{d}(f) = \frac{1}{R(f)} + \theta(f) + \sum_{p=1}^{\infty} \frac{\min\{1, \alpha_p^1(f) + \dots + \alpha_p^4(f)\}}{2^p} + \sum_{p,q=1}^{\infty} \frac{\min\{1, \beta_{p,q}(f)\}}{2^{p+q}}. $$
**Proposition 11.16.** $\mathfrak{D}$ is a distortion of $\mathcal{A}_*/\mathcal{B}$.

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

**Proposition 11.17.** Iso($\mathcal{A}_*$)\$\mathcal{B}^{\text{in}}$ is separable.

Proof.

□

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

**Claim 42.** For $k \in \{1, \dots, \ell\}$, $i \in \mathbb{N}$, $2 \le m \le p$ and $r > 0$ such that $\gamma_{k,m} < r \le 1$, let $P$ be any $r$-extended plaque of $(\xi_{k-1,i}, \lambda_{k-1,i})$ contained in $B_{k-1,m}$. Then $\overline{f_k(P)}$ is contained in some $(r - \gamma_{k,m})$-extended plaque of $(\xi_{k,i}, \lambda_{k,i})$.

Since $R_k \ge p+3$ by (11.6) and (??), $\overline{f_k(P)}$ is contained in the star of some $\frac{1}{2}$-extended plaque of $\lambda_{k,i}$, which in turn is contained in some plaque of $V'_i$. Then this claim follows because

$$ \lambda_{k,i} \circ f_k(y) \ge \lambda_{k-1,i}(y) - |\lambda_{k,i} \circ f_k(y) - \lambda_{k-1,i}(y)| > r - \sigma_{k,m} $$

for all $y \in P$.

**Claim 43.** For all $k \in \{1, \dots, \ell\}$, we have

$$ d_k(x_k, \hat{f}_k(x_0)) \le \frac{1}{1 - \sigma_{1,2} - \alpha_1 - \dots - \sigma_{k,2} - \alpha_k} < 2. $$

The second inequality of this claim holds by (??) and (??), and the first inequality is proved by induction on $k$. Fix any $r_0 \in (\frac{4}{5}, 1)$, and let

$$ r_k = r_0 - \sigma_{1,2} - \alpha_1 - \dots - \sigma_{k,2} - \alpha_k $$

for $k \in \{1, \dots, \ell\}$. Observe that $r_k > \frac{1}{2}$ by (??) and (??). The argument is the same for the cases $k=1$ and $k>1$, but with the following slightly different definitions:

* In the case $k=1$, take some $i \in \mathbb{N}$ and some $r_0$-extended plaque $P$ of $\lambda_{0,i}$ containing $x_0$.

* In the case $k > 1$, assuming that the claim holds for $k-1$, take some $i \in \mathbb{N}$ and some $r_{k-1}$-extended plaque $P$ of $\lambda_{k-1,i}$ such that $x_{k-1}, \hat{f}_{k-1}(x_0) \in P$; this is possible because $1/r_{k-1} < 2$.

Observe that $P \subset B_{k,2}$. Then, since $\sigma_{k,2} < r_{k-1}$, $f_k(P)$ is contained in some $(r_{k-1} - \sigma_{k,2})$-extended plaque $Q_1$ of $\lambda_{k,i}$ by Claim 42; in particular, $f_k(x_{k-1}), \hat{f}_k(x_0) \in Q_1$, and thus

$$ \lambda_{k,i} \circ f_k(x_{k-1}), \lambda_{k,i} \circ \hat{f}_k(x_0) > r_{k-1} - \sigma_{k,2}. $$
Since $d_k(x_k, f_k(x_{k-1})) < 2$ by (\ref{eq:2}) and (\ref{eq:3}), the points $x_k$ and $f_k(x_{k-1})$ are in some $\frac{1}{2}$-extended plaque $Q_2$ of $\lambda_{k,j}$ for some $j \in \mathbb{N}$. It follows that $x_k$ is in the same plaque of $U_{k,i}^j$ as $\hat{f}_k(x_0)$ because $f_k(x_{k-1}) \in Q_1 \cap Q_2$. Moreover

$$ 
\begin{aligned}
\lambda_{k,i}(x_k) &\ge \lambda_{k,i} \circ f_k(x_{k-1}) - |\lambda_{k,i}(x_k) - \lambda_{k,i} \circ f_k(x_{k-1})| \\
&> r_{k-1} - \sigma_{k,2} - \alpha_k \\
&= r_k.
\end{aligned}
$$

We get that $x_k$ and $\hat{f}_k(x_0)$ are in the same $r_k$-extended plaque of $\lambda_{k,i}$, yielding

$$d_k(x_k, \hat{f}_k(x_0)) < \frac{1}{r_k}.$$

This gives the first inequality of the claim because $r_0$ is arbitrary number in $(\frac{4}{5}, 1)$.

**Claim 44.** $B_{0,p} \subset \text{dom } \hat{f}_k$ and $\hat{f}_k(B_{0,p}) \subset B_{k,p+3}$ for all $k \in \{1, \dots, \ell\}$.

Again, we proceed by induction on $k$ to prove this assertion, which is obvious for $k=1$. Then suppose that $k > 1$ and that this property holds for $k-1$. Since $R_k > p+3$ by (11.6) and (\ref{eq:3}), we get $B_{0,p} \subset \text{dom } \hat{f}_k$. Now, take any $x \in B_{0,p}$. We have

$$\frac{1}{r_1} + \dots + \frac{1}{r_q} < p$$

for some finite sequence $(r_1, \dots, r_q)$ in $(0,1)$ such that, for each $a \in \{1, \dots, q\}$, there is some $i_a \in \mathbb{N}$ and some $r_a$-extended plaque $P_a$ of $\lambda_{0,i_a}$ so that $x_0 \in P_1, x \in P_q$ and $P_{a-1} \cap P_a \neq \emptyset$ if $a > 1$. Notice that each $P_a$ is contained in $B_{0,p}$. Let

$$s_{k,a} = r_a - \sigma_{1,p+3} - \dots - \sigma_{k,p+3},$$

which is positive by (11.6) and (\ref{eq:3}). By applying Claim 42 inductively on $k$, it follows that there is an $s_{k,a}$-extended plaque $Q_{k,a}$ of $\lambda_{k,i_a}$ that contains $\hat{f}_k(P_a)$. Moreover

$$ 
\begin{aligned}
\hat{f}_k(x_0) \in \hat{f}_k(P_1) \subset Q_{k,1}, \quad \hat{f}_k(x) \in \hat{f}_k(P_q) \subset Q_{k,q}, \\
Q_{k,a-1} \cap Q_{k,a} \supset \hat{f}_k(P_{a-1} \cap P_a) \neq \emptyset
\end{aligned}
$$

if $a > 1$, yielding

$$d_k(\hat{f}_k(x_0), \hat{f}_k(x)) \le \frac{1}{s_{k,1}} + \dots + \frac{1}{s_{k,q}} < p+1$$

by (11.6), (\ref{eq:3}) and (\ref{eq:4}). Then, from Claim 43 and the triangle inequality, it follows that $d_k(x_k, \hat{f}_k(x)) < p+3$; i.e., $\hat{f}_k(x) \in B_{k,p+3}$.

**Claim 45.** $\hat{R}_k \ge p$ for all $k \in \{1, \dots, \ell\}$.
We know that $B_{0,p} \subset \text{dom } \hat{f}_k$ by Claim 44. Let $P$ be a $\frac{1}{2}$-extended plaque of any $\lambda_{0,i}$ contained in $B_{0,p}$. Since

$$s_k = \frac{1}{2} - \sigma_{1,p} - \dots - \sigma_{k,p} > 0$$

by (??) and (??). Then, by applying Claim 42 inductively on $k$, it follows that $\overline{\hat{f}_k(P)}$ is contained in some $s_k$-extended plaque of $\lambda_{k,i}$, which in turn is contained in the star of some $\frac{1}{2}$-extended plaque of $\lambda_{k,i}$

**Claim 46.** For all $k \in \{1, \dots, \ell\}$, we have

$$(11.22) \qquad \hat{\sigma}_{k,p} \le \sigma_{1,p+3} + \dots + \sigma_{k,p+3},$$

$$(11.23) \qquad \hat{\tau}_{k,p} \le \tau_{1,p+3} + \dots + \tau_{k,p+3},$$

$$(11.24) \qquad \hat{\theta}_{k,p} \le \theta_{1,p+3} + \dots + \theta_{k,p+3},$$

$$(11.25) \qquad \hat{\alpha}_k \le \alpha_1 + \dots + \alpha_k + 2(\sigma_{2,2} + \dots + \sigma_{k,2})$$

$$(11.26) \qquad \hat{\beta}_k \le \beta_1 + \dots + \beta_k + 2(\tau_{2,2} + \dots + \tau_{k,2}).$$

These inequalities are obvious for the case $k=1$, and we proceed by induction on $k$. Take any $k > 1$ so that these inequalities hold for $k-1$, and fix any $x \in B_{0,p}$. Since $\hat{f}_{k-1}(x) \in B_{k-1,p+3}$ by Claim 44, we get the following estimates:

$$\begin{aligned} |\lambda_{k,i} \circ \hat{f}_k(x) - \lambda_{0,i}(x)| &\le |\lambda_{k,i} \circ \hat{f}_k(x) - \lambda_{k-1,i} \circ \hat{f}_{k-1}(x)| + |\lambda_{k-1,i} \circ \hat{f}_{k-1}(x) - \lambda_{0,i}(x)| \\ &\le \sigma_{1,p+3} + \dots + \sigma_{k,p+3}, \end{aligned}$$

which shows (11.22);

$$\begin{aligned} \|\lambda_{k,i} \circ \hat{f}_k(x) \cdot \xi_{k,i} \circ \hat{f}_k(x) - \lambda_{0,i}(x) \cdot \xi_{0,i}(x)\| &\le \|\lambda_{k,i} \circ \hat{f}_k(x) \cdot \xi_{k,i} \circ \hat{f}_k(x) - \lambda_{k-1,i} \circ \hat{f}_{k-1}(x) \cdot \xi_{k-1,i} \circ \hat{f}_{k-1}(x)\| \\ &\quad + \|\lambda_{k-1,i} \circ \hat{f}_{k-1}(x) \cdot \xi_{k-1,i} \circ \hat{f}_{k-1}(x) - \lambda_{0,i}(x) \cdot \xi_{0,i}(x)\| \\ &\le \tau_{1,p+3} + \dots + \tau_{k,p+3}, \end{aligned}$$

yielding (11.23);

$$\begin{aligned} |\lambda_{k,i} \circ \hat{f}_k(x) \cdot u_{k,i,j} \circ \hat{f}_k(x) - \lambda_{0,i}(x) \cdot u_{0,i,j}(x)| &\le |\lambda_{k,i} \circ \hat{f}_k(x) \cdot u_{k,i,j} \circ \hat{f}_k(x) - \lambda_{k-1,i} \circ \hat{f}_{k-1}(x) \cdot u_{k-1,i,j} \circ \hat{f}_{k-1}(x)| \\ &\quad + |\lambda_{k-1,i} \circ \hat{f}_{k-1}(x) \cdot u_{k-1,i,j} \circ \hat{f}_{k-1}(x) - \lambda_{0,i}(x) \cdot u_{0,i,j}(x)| \\ &\le \theta_{1,p+3} + \dots + \theta_{k,p+3}, \end{aligned}$$
which proves (11.24);

$$
\begin{aligned}
|\lambda_{k,i} \circ \hat{f}_k(x_0) - \lambda_{k,i}(x_k)| &\le |\lambda_{k,i} \circ \hat{f}_k(x_0) - \lambda_{k-1,i} \circ \hat{f}_{k-1}(x_0)| \\
&\quad + |\lambda_{k-1,i} \circ \hat{f}_{k-1}(x_0) - \lambda_{k-1,i}(x_{k-1})| \\
&\quad + |\lambda_{k-1,i}(x_{k-1}) - \lambda_{k,i} \circ f_k(x_{k-1})| \\
&\quad + |\lambda_{k,i} \circ f_k(x_{k-1}) - \lambda_{k,i}(x_k)| \\
&\le \alpha_1 + \dots + \alpha_k + 2(\sigma_{2,2} + \dots + \sigma_{k,2})
\end{aligned}
$$

by Claim 43, yielding (11.25); and

$$
\begin{aligned}
\|\lambda_{k,i} \circ \hat{f}_k(x_0) \cdot \xi_{k,i} \circ \hat{f}_k(x_0) - \lambda_{k,i}(x_k) \cdot \xi_{k,i}(x_k)\| &\le \|\lambda_{k,i} \circ \hat{f}_k(x_0) \cdot \xi_{k,i} \circ \hat{f}_k(x_0) - \lambda_{k-1,i} \circ \hat{f}_{k-1}(x_0) \cdot \xi_{k-1,i} \circ \hat{f}_{k-1}(x_0)\| \\
&\quad + \|\lambda_{k-1,i} \circ \hat{f}_{k-1}(x_0) \cdot \xi_{k-1,i} \circ \hat{f}_{k-1}(x_0) - \lambda_{k-1,i}(x_{k-1}) \cdot \xi_{k-1,i}(x_{k-1})\| \\
&\quad + \|\lambda_{k-1,i}(x_{k-1}) \cdot \xi_{k-1,i}(x_{k-1}) - \lambda_{k,i} \circ f_k(x_{k-1}) \cdot \xi_{k,i} \circ f_k(x_{k-1})\| \\
&\quad + \|\lambda_{k,i} \circ f_k(x_{k-1}) \cdot \xi_{k,i} \circ f_k(x_{k-1}) - \lambda_{k,i}(x_k) \cdot \xi_{k,i}(x_k)\| \\
&\le \beta_1 + \dots + \beta_k + 2(\tau_{2,2} + \dots + \tau_{k,2})
\end{aligned}
$$

by Claim 43, showing (11.26).

11.2 Generalized Gromov space

We need a new Gromov-Hausdorff distance between pointed metric spaces with distinguished
closed subsets, which is defined by

$$
d_{GH}((M, A, x), (N, B, y)) = \inf\{d_H(M, N) + d_H(A, B) + d(x, y)\},
$$

where, once more, all admissible metrics on M × N are considered. Then the *generalized Gromov space* $\hat{\mathcal{M}}_*$ is the collection of isometry classes $[M, A, x]$ of proper pointed metric spaces with distinguished closed subsets, $(M, A, x)$, endowed with the *generalized Gromov topology*, which is defined by the following convergence condition: $[M_i, A_i, x_i] \to [M, A, x]$ in $\hat{\mathcal{M}}_*$ if the generalized Gromov-Hausdorff distance between the triples

$$
(\overline{B}_{M_i}(x_i, r), A_i \cap \overline{B}_{M_i}(x_i, r), x_i), \quad (\overline{B}_M(x, r), A \cap \overline{B}_M(x, r), x)
$$

converges to zero for each $r > 0$.

Now, for each proper metric space with a distinguished closed subset, (M, A), there is a
canonical continuous map ̂l<sub>(M,A)</sub> : M → ̂M<sub>*</sub> defined by ̂l<sub>(M,A)</sub>(x) = [M, A, x]. All general
properties indicated for M<sub>*</sub> have corresponding versions for ̂M<sub>*</sub>.

The spaces $\mathcal{R}_*(n)$ and $\mathcal{R}_*^{m,\alpha}(n)$ also have versions $\hat{\mathcal{R}}_*(n)$ and $\hat{\mathcal{R}}_*^{m,\alpha}(n)$ when distinguished closed subsets are considered. Furthermore the indicated compactness properties of complete connected Riemannian manifolds of bounded geometry can be generalized to $\hat{\mathcal{M}}_*$ and $\hat{\mathcal{R}}_*^{m,\alpha}(n)$ by considering distinguished closed subsets. In fact, with some work, the proofs are reduced to the corresponding properties for $\mathcal{M}_*$ and $\hat{\mathcal{R}}_*^{m,\alpha}(n)$, and the fact that the family of closed subsets of a compact metric space is compact with the Hausdorff distance.
## 11.3 Ideas of the proofs

Let $M$ be a complete connected Riemannian manifold of bounded geometry.

To prove Lemma ??, we take any separating net $A_0$ in $M$. Then, to construct a closed subset $A \subset M$ so that $(M, A)$ is aperiodic and aperiodic at infinity, it is enough to take closed balls of appropriate different radius centered at each point of $A_0$.

To prove Lemma ??, we take an exausting sequence of compact domains $D_k$ in $M$, and corresponding “copies” $D_{k,l}$ of each $D_k$ defining a net in $M$. Again choose any separating net $A_0$ in $M$. Then define a sequence of separating nets $A_k$ inductively by setting that $A_{k+1} \cap D_{k,l}$ is equal to the corresponding “copy” of $A_k \cap D_k$. This sequence stabilizes on compact subsets, defining a separating net $A_\infty$ at the limit. This $A_\infty$ is quasi-repetitive and, to remove its possible periodicity and periodicity at infinity, we use the above trick of taking closed balls of the apropriate radius. This defines a closed subset $A$ so that $(M, A)$ is quasi-repetitive, aperiodic and aperiodic at infinity.

Now Theorems ?? and ?? follow easily from Lemmas ?? and ?? by using the properties indicated in Section 11.2.

Finally, Theorem ?? can be proved as follows. By the properties of $L$, the canonical map $\iota: X \to \mathcal{M}_*$ is injective. Moreover it is continuous because $X = X_0$. So $\iota: X \to \iota(X)$ is a homeomorphism since $X$ is compact, and thus $X$ is isometric to $\iota(X)$ as Riemannian foliated spaces. The result now follows easily

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

$$ \frac{1}{r_1} + \cdots + \frac{1}{r_q} < p \implies \frac{1}{r_1 - 2^{p+7}\delta} + \cdots + \frac{1}{r_q - 2^{p+7}\delta} < p + 1 $$

for any finite sequence $(r_1, \dots, r_m)$ in $(0, 1)$ of arbitrary length $m$; necessarily, $q < p$ and $r_a > 1/p$ for all $a \in \{1, \dots, q\}$

*****************************
Chapter 12

Dynamics in Gromov spaces

**12.1 Compactness of classes of metric spaces**

**Definition 12.1.** For a compact metric space $X$, let $\text{Cov}(X, \varepsilon)$ denote the minimum number of balls of radius $\varepsilon$ needed to cover $X$. This function $\text{Cov}(X, \varepsilon)$ is called the covering function of $X$.

Two elementary but important observations are noted in the following lemma.

**Lemma 12.2.** Let $X$ be a compact metric space.

(1) If $\text{Cov}(X, \varepsilon) \le N$, then there is a finite metric space $Y$, of cardinality at most $N$, such that $d_{GH}(X, Y) \le \varepsilon$.

(2) If $d_{GH}(X, Y) < d$, then $\text{Cov}(X, \varepsilon + 2d) \le \text{Cov}(Y, \varepsilon)$.

The following result is due to Gromov and characterizes precompact classes of compact metric spaces.

**Lemma 12.3.** For a class $\mathcal{C}$ of compact metric spaces, the following statements are equivalent:

(1) $\mathcal{C}$ is precompact.

(2) There is a function $C : (0, \infty) \to (0, \infty)$ such that $\text{Cov}(X, \varepsilon) \le C(\varepsilon)$ for all $X$ in $\mathcal{C}$.

*Proof.* If $\mathcal{C}$ is precompact, then for any $\varepsilon > 0$ there are metric spaces $X_1, \cdots, X_k$ in $\mathcal{C}$ that form an $\varepsilon/4$-net in $\mathcal{C}$ for the Gromov-Hausdorff distance. Let $C(\varepsilon) = \max_{i=1}^k \text{Cov}(X_i, \varepsilon/2)$. Given $X$ in $\mathcal{C}$, choose $X_i$ in the above family such that $d_{GH}(X, X_i) < \varepsilon/4$ and obtain that $\text{Cov}(X, \varepsilon) \le \text{Cov}(X_i, \varepsilon/2) \le C(\varepsilon)$.

Every $X$ in $\mathcal{C}$ is within $\varepsilon$ of some finite metric space with cardinality at most $C(\varepsilon)$. Since also $\text{diam} X \le \varepsilon C(\varepsilon/2)$, it may be assumed that the distance between any two points in such finite metric space is at most $\varepsilon C(\varepsilon/2)$. Each such finite metric space corresponds to an equivalence class of symmetric, zero-diagonal, matrix of order $C(\varepsilon)$ and with entries $d_{ij}$ in
the interval $[0, C(\varepsilon/2)]$ satisfying $d_{ij} \le d_{ik} + d_{kj}$, where two such matrices are equivalent if they are permutations of each other. The collection of these equivalence classes of matrices is compact, so it follows that there is a finite collection of finite metric spaces which is an $\varepsilon/2$-net in the Gromov-Hausdorff norm. $\square$

This lemma has the following consequence in the context of pointed metric spaces.

**Corollary 12.4.** A collection $\mathcal{C}$ in $M_*$ is precompact if and only if for each $R > 0$, the collection of balls $\{B(x, R) | x \in X, X \in \mathcal{C}\}$ is precompact in the Gromov-Hausdorff space.

**Lemma 12.5.** Let $C : (0, \alpha) \to (0, \infty)$ be continuous, and let $\mathcal{C}$ be the collection of compact metric spaces such that $\mathrm{Cov}(X, \varepsilon) \le C(\varepsilon)$ for all $X$ in $\mathcal{C}$ and all $\varepsilon > 0$. Then $\mathcal{C}$ is compact.

If the metric space $X$ is not compact then the covering function $\mathrm{Cov}(X, \varepsilon)$ is not defined. The following definition resolves this inconvenience.

**Definition 12.6.** A metric space $X$ has finite metric covering dimension if there is a function $C : (0, \infty) \to (0, \infty)$ such that, for every $\varepsilon > 0$, every ball of radius $R$ in $X$ can be covered by at most $C(\varepsilon)$ balls of radius $\varepsilon R$, for all $R > 0$.

Equivalently, $X$ has finite metric covering dimension if there is a function $C(\varepsilon)$ such that, for every $x$ in $X$, the covering function of the closed ball $\bar{B}(x, R)$ in $X$ satisfies $\mathrm{Cov}(\bar{B}(x, R), \varepsilon R) \le C(\varepsilon)$, for all $R > 0$ and $\varepsilon > 0$.

**Remark.** To show that $X$ has finite metric covering dimension, it suffices to find a function $C$ defined on $(0, \alpha]$, for some $\alpha > 0$, and satisfying $\mathrm{Cov}(B(x, R), \varepsilon R) \le C(\varepsilon)$ for all $\varepsilon \le \alpha$, because then the function $\tilde{C}$ given by $\tilde{C}(\varepsilon) = C(\varepsilon)$ if $\varepsilon \le \alpha$ and $\tilde{C}(\varepsilon) = C(\alpha)$ if $\varepsilon > \alpha$, satisfies $\mathrm{Cov}(B(x, R), \varepsilon R) \le \tilde{C}(\varepsilon)$ for all $\varepsilon > 0$.

More generally, to show that $X$ has finite metric covering dimension, it suffices to show that there is a constant $C$ such that, for all $R > 0$, every ball of radius $R$ can be covered with at most $C$ ball of radius $R/2$.

**Remark.** This concept of metric covering dimension originates with Kolmogoroff and Ti-homiroff, at least in the context of compact metric spaces. In the present setting of noncompact spaces, it was studied in detail by Assouad [?], and it is sometimes called the Assouad dimension.

If $X$ has finite metric covering dimension, then the infimum of all $\beta > 0$ such that $\mathrm{Cov}(B(x, R), \varepsilon R) \le C\varepsilon^{-\beta}$ for all $R > 0$ and all $x \in X$, in some interval $(0, \alpha]$, is called the metric covering dimension of $X$.

**Remark.** The Hausdorff dimension of a metric space does not exceed its metric covering dimension.

For the work in the next section it will be more appropriate to use a localized version of the concept of metric covering dimension. This is defined as follows.
**Definition 12.7.** A metric space $X$ has finite metric covering dimension at $x$ if there is a function $C_x = C_x(\varepsilon)$ such that for every $\varepsilon > 0$ and every $R > 0$, the closed ball $B(x, R)$ can be covered by at most $C_x(\varepsilon)$ balls of radius $\varepsilon R$.

The function $C_x$ can always be taken to be of the form $C_x(\varepsilon) = C\varepsilon^{-\beta}$, for some $\beta > 0$. The smallest such $\beta$ is called the metric dimension of $X$ at $x$, and denoted by $\beta_x$. The metric dimension at $x$ can also be defined by

$$ \beta_x = \limsup_{\varepsilon \to 0} \limsup_{R \to \infty} \frac{\mathrm{Cov}(B(x, R), \varepsilon R)}{-\log \varepsilon}. $$

It is clear that if $X$ has finite metric dimension at one point then it has also finite metric covering dimension at any other point. However the function $C_x$ is not necessarily the same as the function $C_y$, much like the case of the growth function of a riemannian manifold. Indeed, if the ball of center $x$ and radius $R$ can be covered by at least $C_x(\varepsilon)$ balls of radius $\varepsilon R$, then the ball of radius $R$ and center $y$ can be covered by at least $C_x(\varepsilon R/R+d(x, y))$ balls of radius $\varepsilon R$. Thus if $C_x(\varepsilon) \le C_x\varepsilon^{-\beta_x}$, then $C_y(\varepsilon) \le C_x D\varepsilon^{-\beta_x}$, where $D$ is a constant that depends on the distance $d(x, y)$. (This assumes that the metric space has local homogeneous geometry: that all the balls or radius 1 are uniformly quasi-isometric.) They could be taken to be the same if the space has finite metric covering dimension.

**Example 12.8.** If the metric space has finite metric covering dimension at one point and is quasi-symmetric, then it has finite metric covering dimension.

**Proposition 12.9.** If $X$ is a Riemannian manifold of bounded geometry and has finite metric covering dimension, then it has polynomial growth.

*Proof.* The function $C$ can always be taken to be of the form $C(\varepsilon) = C\varepsilon^{-\beta}$, for some constant $C$. Therefore, the ball $B(x, R)$ can be covered with at most $CR^\beta$ balls of radius 1. Because $X$ is also of bounded geometry, there are upper and lower bounds for the volume of all balls in $X$ of radius 1, thus the result. $\square$

**Remark.** The converse is not necessarily true, although it is true if $X$ is quasi-symmetric.

**Proposition 12.10.** If $X$ is a Riemannian manifold of bounded geometry, of polynomial growth, and quasi-symmetric, then $X$ has finite metric covering dimension,

*Proof.* Suppose that the volume of the ball $B(x, R)$ is bounded above by $R^k$, for all $R > 0$ and all $x$ in $X$. $\square$

**Definition 12.11.** For a metric space $X$ with distance function $d$ and for a positive real number $\lambda$, let $X_\lambda$ denote the metric space with underlying set $X$ and distance function $d_\lambda$ given by $d_\lambda(x, y) = \lambda d(x, y)$.

**Lemma 12.12.** If $X$ is a metric space of finite metric covering dimension and $\lambda > 0$, then the metric space $\lambda X$ also has finite metric covering dimension. In fact, the function $C$ that bounds the covering function of $X$ also bounds the covering function of $X_\lambda$.
*Proof.* The ball of radius $R$ and center $x$ in $X_\lambda$ is, as a set, the ball $B(x, R/\lambda)$ in $X$. Given $\varepsilon > 0$, this ball $B(x, R/\lambda)$ can be covered by at most $C(\varepsilon)$ balls in $X$ of radius $\varepsilon R/\lambda$, hence the ball of radius $R$ in $X_\lambda$ can be covered by at most $C(\varepsilon)$ ball of radius $\varepsilon R$ in $X_\lambda$. $\square$

**Proposition 12.13.** Let $X$ be a metric space. Then the family of metric spaces $X_\lambda$, $0 < \lambda \le 1$, is *precompact* if and only if $X$ has finite metric covering dimension.

*Proof.* The statement follows from Lemma ?? and Lemma 12.3. $\square$

As a consequence, if $C$ is a continuous function, then the collection of metric classes with covering function dominated by $C$ is both compact and invariant under the action of the positive real numbers on $M_*$ given by $[X, x] \mapsto [X_\lambda, x]$.

**Lemma 12.14.** For $[M, x]$ and $\lambda \to \infty$, the spaces $[X_\lambda, x]$ converge in the lipschitz sense to the tangent space $[T_x M, 0_x]$.

Note that (as remarked by Gromov in [?], if every ball of radius $0 \le r \le 1$ centered at $x$ can be covered by $R$ balls of radius $r/2$ (for some fixed $R$), then there is a sequence $\lambda_i \to \infty$ such that $[\lambda_i X, x]$ converges in $\mathcal{E}_*$ to a proper metric space (which can be regarded as the tangent cone of $X$ at $x$).

**Proposition 12.15.** Let $X$ be the universal cover of a metric space whose fundamental group is isomorphic to $Z^n$. For $x$ in $X$, the sequence of spaces $[2^{-k}X, x]$ converges to $[\mathbb{R}^n, 0]$, equipped with a metric arising from a norm.

In general, the limit of the metric classes $[X_\lambda, x]$ as $\lambda \to 0$ may not exist. If it does, then the limit $[X, x]$ is such that $[X, x] \sim [X_\lambda, x]$ for all $\lambda > 0$.

## 12.2 Dynamics

The operation $[X, x] \to [\lambda X, x]$ defined an action on Gromov-Hausdorff space. Let $M_C$ denote the classes of metric spaces with covering dimension $\le C$. This is a compact space.

The action is $t \in (0, \infty) \times [X, x] \mapsto [e^{-t}X, x]$.

**Proposition 12.16.** If $[X, x]$ is a periodic point of the flow $\Phi_\lambda$, and $X$ is topological manifold, then $X$ is homeomorphic to Euclidean space.

*Proof.* If $[\lambda X, x] = [X, x]$ for some $\lambda < 1$, then there is a homeomorphism $\phi : X \to X$ such that $d(\phi(x), \phi(y)) = \lambda d(x, y)$; that is, $\phi$ is an expanding map of $X$. Since $X$ is locally euclidean, $X$ is then an increasing union of Euclidean balls, and therefore homeomorphic to a Euclidean space. $\square$

If $[X, x]$ is a recurrent point, then for every $\varepsilon > 0$ there exists a positive integer $n$ such that $d_{GH}([X, x], [(1/n)X, x]) < \varepsilon$.

**Definition 12.17.** A metric measure space is a pair $(X, \mu)$ consisting of a metric space $X$ (proper) and a Borel probability measure $\mu$ on $X$.
If $(X, \mu)$ is a metric measure space, then the map $\Phi : x \in X \to [X, x] \in \mathcal{M}_*$ induces a probability measure $\mu = \Phi_*\nu$ on Gromov-Hausdorff space $\mathcal{M}_*$.

*******

**Example 12.18.** In general, the Gromov-Hausdorff distance between metric spaces $[X, x]$ and $[\lambda X, x]$ need not be finite. For example, let $X = \{0, 1, 4, 9, \dots\}$. If the distance $d_{GH}([X, 0], [2X, 0]) < \infty$, then there are mappings $f$ and $g$ from $X$ into $X$ such that:

$$ 
\begin{aligned}
2d(f(x), f(y)) &\le d(x, y) + R, \\
d(g(x), g(y)) &\le 2d(x, y) + R, \\
d(gf(x), x) &\le R, \\
2d(fg(x), x) &\le R,
\end{aligned}
$$

for some $R > 0$.

Because $d(n^2, X \setminus \{n^2\}) \to \infty$ as $n \to \infty$, it follows that $gf(x) = x = fg(x)$ for every sufficiently large $x$ in $X$. Therefore

$$ 
\begin{aligned}
4n + 2 &= 2(n + 1)^2 - 2n^2 \\
&= 2d((n + 1)^2, n^2) \\
&= 2(fg((n + 1)^2), fg(n^2)) \\
&\le d((n + 1)^2, n^2) + R \\
&= (n + 1)^2 - n^2 + R \\
&= 2n + 1 + R
\end{aligned}
$$

for every $n$ sufficiently large, which is a contradiction.

Is there a condition that implies $d_{GH}((X, x), (\lambda X, x)) < \infty$. For instance, there is $R > 0$ such that $d_{GH}((X, x), (X, y)) < R$ for every $x, y \in X$.

*******

Previously the space of all metric spaces was introduced. The Gromov-Hausdorff distance can be used to make it into a topological space. If $X$ is a (isometry class of) metric spaces, then $\mathcal{M}_X = \{Y \mid d_{GH}(X, Y) < \infty\}$ is in fact a complete metric space.

For any $\lambda > 0$, $\phi_\lambda$ maps $\mathcal{M}_X$ into itself. This is so because on the one hand, for any metric spaces $M, N$, the Gromov-Hausdorff distance $d_{GH}(\phi_\lambda M, \phi_\lambda N) = \lambda d_{GH}(M, N)$, and on the other hand, $d_{GH}(M, \phi_\lambda M) < \infty$ by Lemma ??.

It follows that if $\lambda < 1$ then the map $\phi_\lambda : \mathcal{M}_X \to \mathcal{M}_X$ is a contractive mapping and thus it has a unique fixed point. If $X$ denotes that unique fixed point, then $X \sim \lambda X$, and for any $\mu > 0$, $\mu X \sim \mu \lambda X = \lambda \mu X$, so that $\mu X$ is also a fixed point of $\phi_{\lambda X}$. Therefore $X \sim \mu X$ for all $\mu > 0$.

In general, $\Phi_\lambda (\lambda \neq 1)$ does not map $\mathcal{E}_*[X, x]$ into itself. If it does, that is, if $[X, x]$ is such that the Gromov-Hausdorff distance $d_{GH}([X, x], [\lambda_0 X, x]) < \infty$ for some $\lambda_0 \neq 1$, then $\Phi_{\lambda_0}$ maps $\mathcal{E}_*[X, x]$ into itself, as it follows from the triangle inequality applied to the distance
$d_{GH}$. Then it follows that $\Phi_{\lambda_0}$ (or $\Phi_{1/\lambda_0}$ if $\lambda_0 > 1$ is a contraction on $\mathcal{E}_*[X, x]$. This is because the distance

$$d_{GH}([\lambda Y, y], [\lambda Z, z]) = \lambda d_{GH}([Y, y], [Z, z]).$$

Now it is not clear that if $\Phi_{\lambda_0}$ maps $\mathcal{E}_*[X, x]$ into itself, then all the other $\Phi_{\lambda}$ will do the same.

The pointed Gromov-Hausdorff distance between pointed metric spaces $(M, x)$ and $(N, y)$ is defined by

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

where the infimum is taken over all admissible metric $d$ on $M \sqcup N$. The pointed Gromov-Hausdorff distance is not a metric on the family of isometry classes of pointed metric spaces because there are pairs of spaces for their distance is not a finite number. Nevertheless, such “distance” induces a topology on the family of isometry classes of pointed metric spaces, and it will be denoted by $\mathcal{E}_*$.

**Definition 12.19.** Let $f: X \to Y$ be a mapping of pointed metric spaces. The distortion of $f$ is

$$\mathrm{dis} f = \sup_{x,x'} |d_X(x, x') - d_Y(f(x), f(x'))|$$

**Proposition 12.20.** If $(X, x)$ and $(Y, y)$ are pointed metric spaces such that $d_{GH}((X, x), (Y, y)) < \infty$, then there is a mapping of pointed metric spaces $f: X \to Y$ such that $\mathrm{dis} f < \infty$

The pointed Gromov-Hausdorff distance defines an equivalence relation on $\mathcal{E}_*$, whereby two pointed metric spaces are related if their pointed distance is finite. Each equivalence class is then a complete separable metric space.

The natural map $\mathcal{E}_* \to \mathcal{M}_*$ is continuous.

For each proper metric space $M$ there is a canonical map $\kappa_M : M \to \mathcal{E}_*$ defined by $\kappa_M(x) = [M, x]$, and it is clearly continuous*******. As in the case of $\mathcal{M}_*$ and $\iota_M$, the map $\kappa_M$ is injective if and only if the isometry group of $M$ is trivial.

**Lemma 12.21.** If $\lambda > 0$ and $[M, x], [N, y]$ are pointed metric spaces, then the Gromov-Hausdorff distances satisfy

$$d_{GH}([\lambda M, x], [\lambda N, y]) = \lambda d_{GH}([M, x], [N, y])$$

For any pointed metric space $[M, x]$, let $\mathcal{E}_*[M, x]$ denote the subspace of $\mathcal{E}_x$ consisting of all the pointed metric spaces such that $d_{GH}([M, x], [N, y]) < \infty$. Then $\mathcal{E}_*[M, x]$ is a complete metric space.

For $\lambda > 0$, let $\Phi_{\lambda}: \mathcal{E}_*[M, x] \to \mathcal{E}_*[M, x]$ defined by multiplication by $\lambda$, that is

$$\Phi_{\lambda}([N, y]) = [\lambda N, y].$$

It follows from Lemma 12.21 that if $0 < \mu < 1$, then $\Phi_{\mu}$ is a contractive mapping of $\mathcal{E}_*[M, x]$, and thus it has a unique fixed point, by a theorem of Banach. Let $[N_0, y_0]$ denote
this fixed point. Because $\Phi_{\lambda}\Phi_{\mu} = \Phi_{\mu\lambda}$, it follows that $[N, y]_0$ is also a fixed point of any mapping $\Phi_{\lambda}$, for any $\lambda > 0$, and so that $[\lambda N, y] \to [N_0, y_0]$ as $\lambda \to 0$ for any $[N, y]$ in $\mathcal{E}_*[M, x]$.

Suppose that $[M, x]$ is a pointed metric space such that $\Phi_{\lambda}[X, x]$ is also in $\mathcal{E}_*[M, x]$. Then $\Phi_{\lambda}$ maps $\mathcal{E}_*[M, x]$ into itself and thus it has a unique fixed point in $\mathcal{E}_*[M, x]$.

**Proposition 12.22.** For any $[M, x]$, let $H[M, x]$ denote the set of $\lambda$ such that $d_{GH}([\lambda M, x], [M, x]) < \infty$. Then $H[M, x]$ is a closed (multiplicative) subgroup of $(0, \infty)$.

*Proof.* It is strighthforward to verify that $H[M, x]$ is a group, using the properties of the Gromov-Hausdorff distance $d_{GH}$ and Lemma 12.21.

To verify that $H[M, x]$ is a closed subgroup of $(0, \infty)$, note that for $\lambda \neq 1$ in $H[M, x]$ each of the maps $\Phi_{\lambda}$ has a unique fixed point in $\mathcal{E}_*[M, x]$, and because they commute, all the maps $\Phi_{\lambda}$, $\lambda \in H[M, x]$ share the same unique fixed point in $\mathcal{E}_*[M, x]$.

Let $[Y, y]$ denote this fixed point and suppose that $\lambda_n$ is a sequence in $H[M, x]$ and $\lambda_n \to \lambda > 0$. Because $[\lambda_n Y, y] = [Y, y]$, for each $\lambda_n$ there is an isometry $\phi_n : [\lambda_n Y, y] \to [Y, y]$, that is to say, a mapping $\phi_n : Y \to Y$ which fixes $y$ and satisfies $d(\phi_n u, \phi_n v) = \lambda_n d(u, v)$, for all $u, v$ in $Y$. The sequence $\phi_n$ converges uniformly as $n \to \infty$ to a mapping $\phi : Y \to Y$ such that $\phi(y) = y$ and $d(\phi u, \phi v) = \lambda d(u, v)$, showing that $[\lambda Y, y] = [Y, y]$, that is, that $\Phi_{\lambda}$ also fixes $[Y, y]$. This implies in particular that $\Phi_{\lambda}$ maps $\mathcal{E}_*[Y, y]$ into itself, and thus that $\lambda \in H[M, x]$ because $\mathcal{E}_*[M, x] = \mathcal{E}_*[Y, y]$. $\square$

## 12.3 Asymptotic cones

A few basic facts of the Stone-Čech compactification of the natural numbers $\mathbb{N}$ will now be recalled in order to define the asymptotic cone of a metric space. This compactification, $\mathbb{N}^{\beta}$, of $\mathbb{N}$ is completely characterized by the property that every bounded real-valued function on $\mathbb{N}$ admits a unique continuous extension to $\mathbb{N}^{\beta}$. For notational convenience, if $\omega$ is a point in $\mathbb{N}^{\beta} \setminus \mathbb{N}$, and $f$ is a bounded function on $\mathbb{N}$, then the value at $\omega$ of the unique extension of $f$ to $\beta\mathbb{N}$ is denoted by $\lim_{\omega} f$.

There is a construction of $\mathbb{N}^{\beta}$ using filters on $\mathbb{N}$, and the points in $\mathbb{N}^{\beta} \setminus \mathbb{N}$ correspond to non-principal ultrafilters on $\mathbb{N}$. A filter $\omega$ on $\mathbb{N}$ may be interpreted as a $0-1$ valued, finitely additive, non-trivial measure on $\mathbb{N}$. Thus, viewing a point $\omega$ as a measure on $\mathbb{N}$, to say that $\lim_{\omega} f(n) = a$ is equivalent to saying that for every $\varepsilon > 0$ the set of natural numbers $n$ such that $|f(n) - a| < \varepsilon$ is of full measure for $\omega$. The following property will be needed.

**Lemma 12.23.** Let $\omega$ be an ultrafilter on $\mathbb{N}$, and let $a_n$ be a bounded sequence of real numbers. If $\lim_{\omega} a_n = a$, then $\lim_{\omega} a_{n_k} = a$, for any subsequence $a_{n_k}$ of $a_n$.

*Easy!!!!* Suppose that $\lim_{\omega} a_{n_k} = b$. Then for each $\varepsilon > 0$, the set $K_{\varepsilon}$ of integers $n_k$ such that $|a_{n_k} - b| < \varepsilon$ satisfies $\omega(K_{\varepsilon})$. This implies that $\lim_{\omega} a_n = b$, also. $\square$

Let $(X, d)$ be a metric space and let $x_0$ be a fixed, but otherwise arbitrary, point in $X$. The asymptotic cone of $X$ at $x_0$ is constructed as follows. Let $\omega$ be a non-principal ultrafilter in $\mathbb{N}$,
that is, a point in the boundary of the Stone-Čech compactification of $N$. Let $Y$ be the space of maps $f: N \to X$ such that $d(x_0, f(n)) \le C(f)n$, where $C(f)$ is a constant that depends on $f$ only. For every two such functions $f_1$ and $f_2$, define $d_\omega(f_1, f_2) = \lim_\omega \frac{d(f_1(n), f_2(n))}{n}$. This is not a distance because it may vanish at some pairs of distinct points, so let $\text{Con}_\omega(X) = Y/d_\omega$. This is called the asymptotic cone of $X$ at the point $x_0$ based on the principal ultrafilter $\omega$.

The asymptotic cone of a metric space is again a metric space, although in general it is not locally compact.

The asymptotic cone is independent of the choice of base point; a different choice gives an isometric asymptotic cone.

**Lemma 12.24.** Let $X$ be a metric space and $x_0, x_1$ be any two points in $X$. Then the asymptotic cones $\text{Con}(X, x_0)$ and $\text{Con}(X, x_1)$ are isometric.

*Proof.* If $f: N \to X$ satisfies $f(0) = x_0$, let $\Phi(f)(n) = f(n)$ for $n \ne 0$, and $\Phi(f)(0) = x_1$. Then $d(x_1, f(n)) \le d(x_1, x_0) + d(x_0, f(n)) \le C'(f)n$. Furthermore, $d(\Phi(f_1), \Phi(f_2)) = d(f_1, f_2)$ for any pair of functions $f_1$ and $f_2$ as above. The map $\Phi$ then induces an isometry $\Phi: \text{Con}(X, x_0) \to \text{Con}(X, x_1)$, which is obviously surjective. $\square$

**Proposition 12.25.** If $X_1$ and $X_2$ are metric spaces, then a coarse quasi-isometry $F: X_1 \to X_2$ induces a quasi-isometry $F_\omega: \text{Con}_\omega(X_1) \to \text{Con}_\omega(X_2)$ having the same distortion as $F$.

*Proof.* Elementary. $\square$

**Definition 12.26.** A metric space is coarsely connected if it cannot be partitioned into two non-empty subsets which are at distance at least $c$, for some number $c > 0$.

Equivalently, there is a number $c > 0$ such that for any pair of points $x$ and $y$ in the space there is a finite sequence $x_0 = x, x_1, \cdots, x_n = y$ such that $d(x_i, x_{i+1}) \le c$, for all $i = 1, \cdots, n-1$.

If moreover a finite sequence can be found with the property that $\sum_{i=0}^{n-1} d(x_i, x_{i+1}) \le d(x, y) + b$ then the metric space is called coarsely convex.

**Proposition 12.27.** If $f: X \to Y$ is a coarse map between quasi-connected spaces, then $f$ is a quasi-isometry.

*Proof.*

<figure><img src="image_34.png" /></figure>

**Lemma 12.28.** Let $X$ be a geodesic space, let $x_0$ be a point in $X$. The constant sequence at $x_0$, $c_{x_0}$, represents a point in $\text{Con}(X)$. If $R > 0$ and $f$ is point in the closed ball $B'_\omega(c_{x_0}, R)$, then $f$ can be represented by a sequence $f(n)$ such that $d(x_0, f(n)) \le nR$ for all $n$.

*Proof.* If $f: N \to X$ is any sequence representing $f$, then $\lim_\omega \frac{d(f(n), x_0)}{n} \le R$. Let $\gamma_n$ be a geodesic in $X$ joining $x_0$ and $f(n)$, and define $g: N \to X$ by letting $g(n)$ be the point in $\gamma_n$ at distance $nR$ from $x_0$ if $d(x_0, f(n)) > nR$, and setting $g(n) = f(n)$ otherwise. Then, by construction, $d(g(n), x_0) \le nR$ and $\lim_\omega \frac{d(g(n), f(n))}{n} = 0$, so that $f = g$ in $\text{Con}(X)$. $\square$
More generally, if a point in $\text{Con}_\omega(X)$ is represented by a sequence $f: \mathbb{N} \to X$, then any point in $B(f, R)$ can be represented by a sequence $g: \mathbb{N} \to X$ such that $d(f(n), g(n)) \le nR$, for all $n$.

The following two propositions deal with completeness and local compactness of asymptotic cones of metric spaces.

**Proposition 12.29.** The asymptotic cone of a metric space is a complete metric space.

*Proof.* Let $\{f_k\}$ be a cauchy sequence in $\text{Con}(X)$ and represent each $f_k$ as a function $f_k: \mathbb{N} \to X$ satisfying $d(x_0, f_k(n)) \le C_k n$, where $C_k$ is a constant that depends only on $f_k$. Moreover, because a Cauchy sequence in a metric space is always bounded, there is a positive number $R$ such that every $f_k$ satisfies $d_\omega(f_k, c_{x_0}) \le R/2$. Therefore, for each $k$ there is a subset $A_k$ of $\mathbb{N}$ such that $\omega(A_k) = 1$ and $d_n(x_0, f_k(n)) \le R$. By modifying each $f_k$ outside $A_k$ it may be assumed that each $f_k$ is represented by a sequence with the property that $d_n(x_0, f_k(n)) \le R$ for all $k$ and all $n$.

Let $B_0 = \mathbb{N}$ and inductively construct a nested sequence $B_1 \supset B_2 \supset \cdots$ of subsets of $\mathbb{N}$ such that $\bigcap_j B_j = \emptyset$, the measure $\omega(B_j) = 1$ for all $j$, and

$$|d_j(f_p(n), f_q(n)) - d_\omega(f_p, f_q)| < \frac{1}{j}$$

for all $1 \le p, q \le j$ and all $n$ in $B_j$.

Let $g: \mathbb{N} \to X$ be the sequence defined by $g(n) = f_j(n)$ if $n$ is in $B_j \setminus B_{j+1}$. By construction of the representative sequences of the $f_k$'s, the sequence $g$ just defined satisfies $d_n(g(n), x_0) \le R$, so it represents a point in $\text{Con}(X)$.

It will now be shown that $f_k$ converges to $g$ in $\text{Con}(X)$. Let $\varepsilon > 0$. Then there is an integer $N > 0$ such that $d_\omega(f_p, f_q) < \varepsilon/2$ if $p, q \ge N$, and also $1/N < \varepsilon$. If $p \ge N$ and $n \in B_p$, then $n \in B_j \setminus B_{j+1}$ for some $j \ge p$ and so

$$d_n(f_p(n), g(n)) \le \frac{1}{j} + d_\omega(f_p, f_j) \le \varepsilon$$

It follows that $\lim_\omega d_n(f_p(n), g(n)) \le \varepsilon$, or that $d_\omega(f_p, g) \le \varepsilon$ if $p \ge N$. $\square$

<figure><img src="image_35.png" /></figure>

**Proposition 12.30.** Let $X$ be a geodesic metric space which has finite metric covering dimension at a point $x_0$. Then the asymptotic cone of $X$ is a locally compact metric space.

*Proof.* Let $c_{x_0}$ denote the element of $\text{Con}(X)$ represented by the constant sequence at $x_0$, $c_{x_0}(n) = x_0$ for all $n$. It suffices to show that the closed ball $B(c_{x_0}, R)$ is compact, for each $R > 0$.

For each $n$ let $x_k^n$ be a maximal $R/2$-separated set in the ball $B(x_0, nR)$. Let $f_1, \cdots, f_k$ be defined by $f_j(n) = x_j^n$. Then the balls $B(f_j, R/2)$ in $\text{Con}(X)$ cover the ball $B(c_{x_0}, R)$. Suppose that $f$ is a point in $B'_\omega(c_{x_0}, R)$ such that $d_\omega(f_j, f) > R/2$ for $j = 1, \cdots, k$. By Lemma 12.28, it may be assumed that the sequence representing $f$ satisfies $d(x_0, f(n)) \le nR$, for all $n$ in $\mathbb{N}$. There are subsets $D_1, \cdots, D_k$ of $\mathbb{N}$ such that $\omega(D_j) = 1$ and $d_n(f(n), f_j(n)) >$
$R/2$ for all $n$ in $D_j$, $j = 1, \dots, k$. In particular, if $D = \bigcup_{j=1}^k D_j$, then $d_n(f(n), f_j(n)) > R/2$ for all $j = 1, \dots, k$ and all $n$ in $D$. This implies that $f(n)$ is not in the closed ball $B_n(x_0, R)$ for all $n$ in $D$.

It follows by induction that the closed ball of radius $R$ and center $c_{x_0}$ in $\text{Con}(X)$ can be covered by at most $C(\varepsilon)$ balls of radius $\varepsilon R$. Since $\text{Con}(X)$ is complete (by Proposition 12.29), it follows that such ball is compact and thus that $\text{Con}(X)$ is locally compact. $\square$

**Corollary 12.31.** If $X$ has finite metric covering dimension at $x_0$, then the asymptotic cone of $X$ at $x_0$ also has finite metric covering dimension at $c_{x_0}$.

*Proof.* This is implicit in the proof of the previous proposition. In fact, that argument shows that any function that dominates the covering function of $X$ at $x_0$ also dominates the covering function on $\text{Con}_\omega(X)$ at the base point $c_{x_0}$.

If moreover $X$ has finite metric covering dimension, then $\text{Con}(X)$ also has finite metric covering dimension. $\square$

**Proposition 12.32.** Let $X$ be a foliated space such that all its leaves have covering function bounded above by a continuous function $C$. Then the map that assigns to a point $x$ in $X$ the asymptotic cone (based at $x$) of the holonomy cover of the leaf through $x$ is a Borel map into the Gromov-Hausdorff space $\mathcal{M}_*$.

*Needs reworking.* The hypothesis on $X$ implies that the asymptotic cone of each leaf is a locally compact, complete metric space (Proposition 12.30). Therefore the asymptotic cone map $x \in X \to \text{Con}(X, x) \in \mathcal{M}_*$ is well defined.

Let $R, \varepsilon > 0$ and let $[Y, u]$ be a point in $\mathcal{M}_*$. The family of pointed metric spaces $[Z, z]$ such that the pointed Gromov-Hausdorff distance between the closed ball $\bar{B}(z, R)$ and $\bar{B}(y, R)$ is less than a number $\varepsilon$ is a neighborhood $U_{R,\varepsilon}$ of $[Y, y]$ in $\mathcal{M}_*$.

There are $\varepsilon$-dense subsets $A_z = \{z, z_1, \dots, z_{n(z)}\}$ of $\bar{B}(z, R)$ and $B_z = \{y, y_1, \dots, y_{n(z)}\}$ of $\bar{B}(y, R)$ such that

$$|d(z_i, z_j) - d(y_i, y_j)| \le \varepsilon.$$

It seems that we have to assume the following. There is a fixed $\varepsilon$-net in $\bar{B}(y, R)$ and the others are at distance $\le \varepsilon$ from it, via bi-lipschitz maps.

These spaces have finite covering dimension, so the ball of radius $R$ has an $\varepsilon$-net with at most $C(\varepsilon/R)$ elements.

Thus we may assume that $n(z)$ is bounded above independently of $z$, just depending on $R$. $\dots$

Let $x_n$ be a sequence of points in the foliated space $X$ that converges to $x$ in $X$, and suppose that the asymptotic cones of the leaves $F_{x_n}$ based at the point $x_n$ are within $U(R, \varepsilon)$. Each $u \in \bar{B}(c_{x_i}, R)$ can be represented by a map $f_i : \mathbb{N} \to F_{x_i}$ such that $d(x_i, f_i(n)) < nR$. Because the sequence $x_i$ converges to $x$, there are bilipschitz maps $\phi_{n,i} : \bar{B}(x_i, nR) \to F_x$ with $\phi_{n,i}(x_i) = x$ and with distortion constant $C_{n,i} \to 1$ as $i \to \infty$. For each $n$ there exists an index $i_n$ such that $|C_{n,i} - 1| < \frac{1}{n^2R}$ for all $i \ge i_n$; in particular $\phi_{n,i_n}(\bar{B}(x_i, nR)) \subset \bar{B}(x, nR + 1/n)$.
Let $h: \mathbb{N} \to F_x$ be the sequence defined by $h(n) = \phi_{n,i_n}(f_{i_n}(n))$, and let $\tau(u)$ be the point in $\text{Con}(F_x, \omega)$ defined by the sequence $h$.

**Claim 47.** The map $\tau$ is well defined, that is independent of the representative $f_i$ of $\sigma_i(u)$.

*Proof.* Let $f'_i: \mathbb{N} \to F_{x_i}$ be another sequence representing $\sigma_i(u)$ and let $h': \mathbb{N} \to F_x$ be given by $h'(n) = \phi_{n,i_n}(f'_{i_n}(n))$. Then

$$ 
\begin{align*}
\lim_{\omega} \frac{1}{n} d(h(n), h'(n)) &= \lim_{\omega} \frac{1}{n} d(\phi_{n,i_n}(f_{i_n}(n)), \phi_{n,i_n}(f'_{i_n}(n))) \\
&= \lim_{\omega} \frac{C_{n,i_n}}{n} d(f_{i_n}(n), f'_{i_n}(n)) \\
&= 0
\end{align*}
$$

because $C_{n,i_n} \to 1$ as $n \to \infty$, thus $\lim_{\omega} C_{n,i_n} = 1$ also. Therefore $h$ and $h'$ represent the same element in the asymptotic cone $\text{Con}(F_x, x)$, and so $\sigma(u)$ is well defined. $\square$

**Claim 48.** The map $\tau$ satisfies $\tau(y) = c_x$.

*Proof.* Each $\sigma_i(y) = c_{x_i}$, and $c_{x_i}$ can be represented by the constant sequence $f_i(n) = x_i$ for all $n$. Then $h(n) = \phi_{n,i_n} \circ f_i(n) = \phi_{n,i_n}(x_{i_n}) = x$ is the constant sequence $h: \mathbb{N} \to F_x$ representing $c_x$. $\square$

**Claim 49.** The mapping $\tau: B((Y, y), R) \to B((\text{Con}(F_x, x), c_x), R)$ has distortion $\text{dis}\tau \le \varepsilon$.

*Proof.* Let $u$ and $v$ be two elements in $B((Y, y), R)$ and let $f_i, g_i: \mathbb{N} \to F_{x_i}$ be sequences representing $\sigma_i(u)$ and $\sigma_i(v)$, respectively, with $\sigma_i(u)(n)$ and $\sigma_i(v)(n)$ in the closed ball $B'_{F_i}(x_i, nR)$, for all $n$. Let $h, k: \mathbb{N} \to F_x$ be the sequences given by $h(n) = \phi_{n,i_n}(f_{i_n}(n))$ and $k(n) = \phi_{n,i_n}(g_{i_n}(n))$. Then

$$ 
\begin{align*}
d_{\omega}(\tau(u), \tau(v)) &= \lim_{\omega} \frac{1}{n} d(h(n), k(n)) \\
&= \lim_{\omega} \frac{1}{n} d(\phi_{n,i_n}(f_{i_n}(n)), \phi_{n,i_n}(g_{i_n}(n))) \\
&\le \lim_{\omega} \frac{C_{n,i_n}}{n} d(f_{i_n}(n), g_{i_n}(n)) \\
&= \lim_{\omega} \frac{1}{n} d(f_{i_n}(n), g_{i_n}(n))
\end{align*}
$$

and similarly

$$ 
\begin{align*}
d_{\omega}(\tau(u), \tau(v)) &= \lim_{\omega} \frac{1}{n} d(h(n), k(n)) \\
&= \lim_{\omega} \frac{1}{n} d(\phi_{n,i_n}(f_{i_n}(n)), \phi_{n,i_n}(g_{i_n}(n))) \\
&\ge \lim_{\omega} \frac{1}{nC_{n,i_n}} d(f_{i_n}(n), g_{i_n}(n)) \\
&= \lim_{\omega} \frac{1}{n} d(f_{i_n}(n), g_{i_n}(n))
\end{align*}
$$
so that

$$d_{\omega}(\tau(u), \tau(v)) = \lim_{\omega} \frac{1}{n} d(f_{i_n}(n), g_{i_n}(n)).$$

<figure><img src="image_36.png" /></figure>

Because every open ball in $M_*$ is a countable union of subsets of $M_*$ of the form $U(R, \epsilon)$, it follows that the inverse image of an open ball is a countable union of closed sets, hence a Borel set. This suffices to show that the asymptotic cone map is Borel. □

**Theorem 12.33.** Let $X$ be a foliated space which has a dense leaf of finite metric covering dimension. Then there is a residual, saturated subset of $X$ all whose leaves have isometric asymptotic cones.

*Proof.* The existence of a dense leaf with finite metric dimesnion implies that there is a residual subset saturated subset of $X$ all whose leaves also have finite metric covering dimension.

The asymptotic cone map $X \to M_*$ is borel and constant along the leaves of $X$. The result follows from Theorem ??.

<figure><img src="image_37.png" /></figure>

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

## 12.4 The global Gromov space and equicontinuous foliated spaces

The pointed Gromov-Hausdorff distance between pointed metric spaces $(M, x)$ and $(N, y)$ is defined by

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

where the infimum is taken over all admissible metric $d$ on $M \sqcup N$. The pointed Gromov-Hausdorff distance is not a metric on the family of isometry classes of pointed metric spaces because there are pairs of spaces for their distance is not a finite number. Nevertheless, such “distance” induces a topology on the family of isometry classes of pointed metric spaces, and it will be denoted by $\mathcal{E}_*$.

**Lemma 12.34.** Suppose that $(M, x)$ and $(N, y)$ are complete metric spaces such that the Gromov-Hausdorff pointed distance $d_{GH}((M, x), (N, y)) = 0$. Then there is an isometry $M \to N$ which sends $x$ to $y$.

*Proof.* Let $d_k$ be a sequence of metric on $M \sqcup N$ such that $d_k(M, N) + d_k(x, y) < 1/2^k$. Then there are maps (possibly discontinuous) $f_k : M \to N$ and $g_k : N \to M$ such that $d_k(z, f_k(z)) < 1/2^k$ and $d_k(g_k(w), w) < 1/2^k$ for all $z$ in $M$ and all $w$ in $M$, □

<figure><img src="image_38.png" /></figure>

Let $\mathcal{E}_*$ denote the space of isometry classes of pointed complete metric spaces. The Gromov-Hausdorff pointed “distance” $d_{GH}$ does not induce a metric on $\mathcal{E}_*$ because it is quite possible to have $d_{GH}((M, x), (N, y)) = \infty$ for a pair of pointed metric spaces. But if $(M, x)$
is any complete pointed metric space, then $d_{GH}$ induces a metric on the subspace of classes of metric spaces $[N, y]$ with $d_{GH}([M, x], [N, y]) < \infty$.

The pointed Gromov-Hausdorff distance defines an equivalence relation on $\mathcal{E}_*$, whereby two pointed metric spaces are related if their pointed distance is finite. Each equivalence class is then a complete separable metric space.

**Proposition 12.35.** Let $\mathcal{E}_*[M, x]$ denote the collection of complete pointed metric spaces which are at finite Gromov-Hausdorff pointed distance from $[M, x]$. Then $\mathcal{E}_*[M, x]$ is a complete metric space.

*Proof.* Let $[M_k, x_k]$ be a Cauchy sequence in $\mathcal{E}_*$. After passing to a subsquence, if necessary, it may be assumed that $d_{GH}([M_k, x_k], [M_{k+1}, x_{k+1}]) < 1/2^k$

Select metrics $d_{k,k+1}$ on $M_k \sqcup M_{k+1}$ such that the Hausdorff distance $d_H(M_k, M_{k+1}) < 1/2^k$ in $M_k \sqcup M_{k+1}$ with respect to $d_{k,k+1}$, and also $d_{k,k+1}(x_k, x_{k+1}) < 1/2^k$. Define a metric $d_{k,k+j}$ on $M_k \sqcup M_{k+j}$ by setting

$$d_{k,k+j}(z_k, z_{k+j}) = \min_{z_{k+l} \in M_{k+l}} \sum_{l=0}^{j-1} d_{k+l,k+l+1}(x_{k+l}, x_{k+l+1}).$$

This defines a distance $d$ on $\sqcup_k M_k$ with the property that, with respect to $d$, the Hausdorff distance $d_H(M_k, M_{k+j}) < 1/2^{k-1}$ and $d(x_k, x_{k+j}) < 1/2^{k-1}$.

Let $M_c$ be the collection of sequences $(z_k)$ with $z_k \in M_k$ and $d(z_k, z_l) \to 0$ as $k, l \to \infty$. Define an equivalence relation on $M_c$ by setting $(z_k) \sim (w_k)$ if $d(z_k, w_k) \to 0$ as $k \to \infty$ and let $\hat{M}$ denote the quotient space. Then $\hat{M}$ inherits a metric given by $d([z_k], [w_k]) = \lim_k d(z_k, w_k)$.

Note that the sequence $(x_k)$ (where $x_k$ is the selected point of $M_k$) defines an element of $\hat{M}$.

The distance on $\hat{M}$ can be extended to one on $(\sqcup M_k) \sqcup \hat{M}$ by setting

$$d(z_k, (w_j)) = \lim_j d(z_k, w_j)$$

□

**Corollary 12.36.** The natural map $\mathcal{E}_* \to \mathcal{M}_*$ is continuous.

For each proper metric space $M$ there is a canonical map $\kappa_M : M \to \mathcal{E}_*$ defined by $\kappa_M(x) = [M, x]$.

**Lemma 12.37.** The map $\kappa_M : M \to \mathcal{E}_*$ is continuous.

*Proof.* Let $x_n \to x$ in $M$. For each $n$, let $Y = M \times \{0, 1/n\}$ with the metric $d((x, a), (y, b)) = d(x, y) + |a - b|$. The Hausdorff distance between $M \times \{0\}$ and $M \times \{1/n\}$ in $Y$ is $1/n$, and the distance $d(x, x_n) \to 0$ as $n \to \infty$. Therefore, $\kappa_M(x_n) = [M, x_n] \to \kappa_M(x) = [M, x]$ as $n \to \infty$ in $\mathcal{M}_*$. □
As in the case of $\mathcal{M}_*$ and $\iota_M$, the map $\kappa_M$ is injective if and only if the isometry group of $M$ is trivial.

**Lemma 12.38.** The pointed Gromov-Hausdorff distance $d_{GH}((M, x), (N, y)) < \infty$ if and only if

**Corollary 12.39.** If $(M, x)$ is a pointed metric space, then $[\lambda M, x]$ is in $\mathcal{E}_*[M, x]$ for any $\lambda > 0$.

*Proof.* The identity map $\text{id}: M \to \lambda M$ is a $\lambda$-lipschitz map. $\square$

<figure><img src="image_39.png" /></figure>

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

**Definition 12.40.** For a point $x$ and a real number $r > 0$, let $\delta(x, r)$ denote the minimum number of balls of radius $r/2$ needed to cover the ball $B(x, r)$.

The function $\delta$ is the density at $x$. It is in some respect similar to the growth function at $x$, as this one is comparable to the least number of balls of radius 1 needed to cover the ball $B(x, r)$.

**Definition 12.41.** The space $M$ has finite metric covering dimension at $x$ if

$$ \sup_{r>0} \delta(x, r) < \infty $$
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