Proof. By definition for every $\varepsilon> 0$ there is $n_\varepsilon \geq 0$ such that for every $n\geq n_\varepsilon $ there is an equivariant ${\varepsilon }/{2}$ -approximation $(f_n,\varphi _n,\psi _n)$ between $(X_n,x_n,\Gamma _n)$ and $(X,x,\Gamma )$ . We want to define an equivariant $2\varepsilon $ -approximation $(f_n,\varphi _n,\bar \psi _n)$ between $(X_n,x_n,\Gamma _n)$ and $(X,x,\bar \Gamma )$ .

For every $g\in \Sigma _{1}/{\varepsilon }(\bar \Gamma ,x)$ there is a sequence of isometries $g_k\in \Gamma $ such that $g_k \to g$ uniformly on compact subsets of X. In particular, for every $\delta> 0$ there exists $k_\delta \geq 0$ such that if $k\geq k_\delta $ then $d(g_k(y), g(y)) \leq \delta $ for every $y\in \overline {B}(x, {2}/{\varepsilon })$ . Choosing $\delta $ small enough, we have $g_k \in \Sigma _{{2}/{\varepsilon }}(\Gamma , x)$ for $k \geq k_\delta $ . We define $\bar \psi _n(g) := \psi _n(g_{k_\delta })$ . Observe that for every $y_n\in B(x_n,{1}/{\varepsilon })$ we have

$$ \begin{align*} \begin{aligned} d(f_n(\bar\psi(g)y_n), gf_n(y_n)) &= d(f_n(\psi_n(g_{k_\delta})), gf_n(y_n)) \\ &\leq d(f_n(\psi_n(g_{k_\delta})), g_{k_\delta}f_n(y_n)) + d( g_{k_\delta}f_n(y_n), gf_n(y_n)) \\ &\leq \varepsilon + \delta, \end{aligned} \end{align*} $$

where the last inequality follows since $f_n(y_n)\in \overline {B}(x, {2}/{\varepsilon })$ . Taking $\delta \leq \varepsilon $ , we conclude that $(f_n,\varphi _n, \bar \psi _n)$ is the desired equivariant $2\varepsilon $ -approximation and it is defined for all $n\geq n_\varepsilon $ .

Proof. Let $(g_\omega ^k)^{k\in \mathbb {N}}$ be a sequence of isometries of $\Gamma _\omega $ converging to an isometry $g^\infty $ of $X_\omega $ with respect to the uniform convergence on compact subsets. We want to show that $g^\infty $ coincides with the ultralimit of some sequence of admissible isometries $g_n^\infty $ of $X_n$ .

First of all, we can extract a subsequence, denoted again by $(g_\omega ^k)$ , satisfying $d(g_\omega ^k y_\omega , g^{k+1}_\omega y_\omega ) \leq {1}/{2^k}$ for all $y_\omega \in \overline {B}(x_\omega , k)$ , for all $k\in \mathbb {N}$ . Now, for every fixed $k\in \mathbb {N}$ , let $S^k = \lbrace y_\omega ^1,\ldots ,y_\omega ^{N_k}\rbrace $ be a ${1}/{2^{k+2}}$ -dense subset of $B(x_\omega , k)$ . It is finite since $X_\omega $ is proper. If $y_\omega ^i = \omega $ - $\lim y_n^i$ for $i=1,\ldots ,N_k$ , it is clear that the set $S^k_n = \lbrace y_n^1,\ldots , y_n^{N_k}\rbrace $ is a ${1}/{2^{k+1}}$ -dense subset of $B(x_n, k)$ for $\omega $ -a.e. $(n)$ . For every $k\in \mathbb {N}$ we define the set

$$ \begin{align*}A_k = \bigg\lbrace n \in \mathbb{N} \text{ s.t. } d(g_n^jy_n^i, g_n^{j+1}y_n^i) \leq \frac{1}{2^j} \text{ for all } 1\leq j \leq k,\ i = 1,\ldots, N_j\bigg\rbrace.\end{align*} $$

By definition $\omega (A_k)=1$ and $A_{k+1}\subseteq A_{k}$ for every $k\in \mathbb {N}$ . We set $B=\bigcap _{k\in \mathbb {N}} A_k$ . There are two cases.

Case 1: $\omega (B)=1$ . In this case $d(g_n^ky_n^i, g_n^{k+1}y_n^i) \leq {1}/{2^k}$ for all $i=1,\ldots , N_k$ and all $k\in \mathbb {N}$ , $\omega $ -a.s. Then $\omega $ -a.s. we have $d(g_n^ky_n, g_n^{k+1}y_n) \leq {1}/{2^{k-1}}$ for every $k\in \mathbb {N}$ and every $y_n\in B(x_n,k)$ . We set $g_n^\infty := g_n^n \in \Gamma _n$ . This sequence of isometries is admissible and we denote its ultralimit by $g_\omega ^\infty \in \Gamma _\omega $ . Now we fix $y_\omega = \omega $ - $\lim y_n \in X_\omega $ . By definition $d(x_n,y_n)\leq M$ for $\omega $ -a.e. $(n)$ . We have

$$ \begin{align*} \begin{aligned} d(g^\infty y_\omega, g^\infty_\omega y_\omega) \leq d(g^k_\omega y_\omega, g^\infty_\omega y_\omega) + \frac{1}{2^{k}} &= \omega\text{-}\!\lim d(g_n^ky_n, g_n^n y_n) + \frac{1}{2^k} \\ &\leq \sum_{j=0}^{n-k-1}d(g_n^{k+j}y_n, g_n^{k+j+1}y_n) + \frac{1}{2^k} \\ &\leq \sum_{j=k-1}^\infty \frac{1}{2^j} + \frac{1}{2^k} \leq \frac{1}{2^{k-2}} + \frac{1}{2^k} \leq \frac{1}{2^{k-3}} \end{aligned} \end{align*} $$

for all fixed $k\geq M$ . We conclude that $g^\infty y_\omega = g^\infty _\omega y_\omega $ . By the arbitrariness of $y_\omega $ we get $g^\infty = g^\infty _\omega $ that belongs to $\Gamma _\omega $ .

Case 2: $\omega (B)=0$ . Since $A_1 = \bigsqcup _{j=1}^\infty (A_j\setminus A_{j+1}) \sqcup B$ and since $\omega (A_1) = 1$ , we have that $\omega (\bigsqcup _{j=1}^\infty (A_j\setminus A_{j+1})) = 1$ . We set $C = \bigsqcup _{j=1}^\infty (A_j\setminus A_{j+1})$ . For all $n \in C$ we set $g_n^\infty := g_n^{j(n)}$ , where $j(n)$ is the unique $j\geq 1$ such that $n\in A_j\setminus A_{j+1}$ . We claim that the corresponding ultralimit isometry $g_\omega ^\infty \in \Gamma _\omega $ equals $g^\infty $ . Indeed, let $y_\omega = \omega $ - $\lim y_n \in X_\omega $ . For every fixed $k\in \mathbb {N}$ , consider the set $C_k = \bigsqcup _{j=k}^\infty (A_j\setminus A_{j+1})$ . Each of the sets $A_j\setminus A_{j+1}$ has $\omega $ -measure $0$ , so $\omega (C_k) = 1$ for every fixed k. Let $n\in C_k \subset C$ . By definition $j(n)\geq k$ since $n\in C_k$ . Since $A_{j(n)}\subseteq A_{j(n)-1} \subseteq \cdots \subseteq A_k$ we have

$$ \begin{align*} \begin{aligned} d(g_n^{j(n)}y_n, g_n^k y_n) \leq 2\cdot\frac{1}{2^{k+2}} + \sum_{m = k}^{j(n)} \frac{1}{2^m} \leq \frac{1}{2^{k+1}}\cdot \frac{1}{2^{k-1}} \leq \frac{1}{2^{k-2}} \end{aligned} \end{align*} $$

as soon as $k> d(y_\omega , x_\omega )$ and $n\in C_k$ . The set $C_k$ has $\omega $ -measure $1$ , so we conclude that $d(g_\omega ^\infty y_\omega , g^\infty y_\omega ) \leq {1}/{2^{k-2}}$ . By the arbitrariness of k and $y_\omega $ we finally get $g^\infty = g_\omega ^\infty \in \Gamma _\omega $ .

Proof. Let $(X_n,x_n,\Gamma _n) = (X,x,\Gamma )$ for every n and let $\omega $ be a non-principal ultrafilter. By [Reference Cavallucci and SambusettiCS21, Proposition A.3] the map $\iota \colon (X,x) \to (X_\omega ,x_\omega )$ that sends each point y to the ultralimit point corresponding to the constant sequence $y_n=y$ is an isometry sending x to  $x_\omega $ . This means that each point $y_\omega $ of $X_\omega $ can be written as $y_\omega = \omega $ - $\lim y$ for some $y\in X$ . We need to show that $\iota _*\Gamma = \Gamma _\omega $ . We have $\iota _*g(\omega $ - $\lim y) = \omega $ - $\lim gy$ for every $g\in \Gamma $ and $y_\omega = \omega $ - $\lim y \in X_\omega $ , that is, $\iota _*g$ coincides with the ultralimit of the constant sequence $g_n = g$ . In particular, $\iota _*\Gamma \subseteq \Gamma _\omega $ . Now we take $g_\omega = \omega $ - $\lim g_n \in \Gamma _\omega $ , where $g_n\in \Gamma $ is an admissible sequence, that is, $d(x,g_nx)\leq M$ for some M. The set $\Sigma _M(\Gamma , x)$ is compact by the Arzelà–Ascoli theorem [Reference KelleyKel17, Ch. 7 and Theorem 17] since $\Gamma $ is closed, so by [Reference Druţu and KapovichDK18, Lemma 10.25] there exists $g\in \Sigma _M(\Gamma , x)$ such that for every $\varepsilon> 0$ and $R\geq 0$ the set

$$ \begin{align*}\lbrace n \in \mathbb{N} \text{ s.t. } d(g_ny,gy)<\varepsilon \text{ for all } y \in \overline{B}(x,R) \rbrace\end{align*} $$

belongs to $\omega $ . It is clear that the constant sequence g defines the ultralimit isometry $g_\omega $ , that is, $\iota _*(g) = g_\omega $ . Since $g\in \Gamma $ we conclude that $\iota _*\Gamma = \Gamma _\omega $ .

Proof. We divide the proof into steps.

Good definition. Given a point $y_\omega = \omega $ - $\lim y_n \in X_\omega $ , by definition there exists $M\geq 0$ such that $d(x_n,y_n)\leq M \omega $ -a.s. For sufficiently large n the map $f_n$ is defined on $y_n$ and it satisfies $d(f_n(y_n), f_n(x_n)) \leq M + 1$ and $f_n(x_n)=x$ . Then the sequence $(f_n(y_n))$ is $\omega $ -a.s. bounded and $\iota ^{-1}(\omega $ - $\lim f_n(y_n))$ is a well-defined point of X.

Now suppose $(y_n')$ is another sequence such that $\omega $ - $\lim d(y_n,y_n') = 0$ . For every $\varepsilon> 0$ we have $d(y_n,y_n')<\varepsilon \omega $ -a.s. Moreover, arguing as before, $d(f_n(y_n), f_n(y_n')) \leq d(y_n,y_n') + \varepsilon \leq 2\varepsilon \omega $ -a.s. By the arbitrariness of $\varepsilon> 0$ we get $d(\omega $ - $\lim f_n(y_n), \omega $ - $\lim f_n(y_n')) = 0$ . In particular, F is well defined.

Isometric embedding. We fix $y_\omega = \omega $ - $\lim y_n, z_\omega = \omega $ - $\lim z_n \in X_\omega $ and $\varepsilon> 0$ . As usual, all the conditions

$$ \begin{align*} &d(\iota^{-1}(\omega\text{-}\!\lim f_n(y_n)), f_n(y_n)) < \varepsilon, \quad d(\iota^{-1}(\omega\text{-}\!\lim f_n(z_n)), f_n(z_n)) < \varepsilon,\\ &\vert d(y_\omega, z_\omega) - d(y_n,z_n)\vert <\varepsilon, \quad\vert d(f_n(y_n), f_n(z_n)) - d(y_n,z_n) \vert < \varepsilon \end{align*} $$

hold $\omega $ -a.s. Therefore

$$ \begin{align*}\vert d(F(y_\omega), F(z_\omega)) - d(y_\omega, z_\omega) \vert < 4\varepsilon.\end{align*} $$

By the arbitrariness of $\varepsilon $ we conclude that F is an isometric embedding.

Surjectivity. We fix $y\in X$ , $\varepsilon> 0$ and we set $L:= d(x,y)$ . By definition there exists $y_n \in X_n$ such that $d(f_n(y_n),y)<\varepsilon \omega $ -a.s. The sequence $y_n$ is clearly admissible and defines a point $y_\omega = \omega $ - $\lim y_n$ of $X_\omega $ . Since $d(f_n(y_n), y) < \varepsilon \omega $ -a.s., we have that $F(y_\omega ) = \omega $ - $\lim f_n(y_n)$ satisfies $d(F(y_\omega ), y) < 2\varepsilon $ . This shows that y belongs to the closure of $F(X_\omega )$ . Every ultralimit space is a complete metric space [Reference Druţu and KapovichDK18, Corollary 10.64], so F is a closed map. Indeed, if $F(y_\omega ^k)$ is a convergent sequence, then it is Cauchy. Since F is an isometric embedding, the sequence $(y_\omega ^k)$ is Cauchy and therefore converges. By continuity of F we conclude that the limit point of the sequence $F(y_\omega ^k)$ belongs to the image of F. Hence $F(X_\omega )$ is closed and $y\in F(X_\omega )$ , showing that F is surjective.

F is equivariant. It is clear that $F(x_\omega ) = \iota ^{-1}(\omega $ - $\lim f_n(x_n)) = x$ . It remains to show that $F_*(\Gamma _\omega ) = \Gamma $ . We take $g_\omega = \omega $ - $\lim g_n \in \Gamma _\omega $ . Then $F_*(g_\omega )$ acts on the point y of X as $F_* g_\omega (y) = F\circ g_\omega \circ F^{-1}(y)$ . Clearly $F^{-1}(y) = \omega $ - $\lim y_n$ , where $y_n$ is any sequence such that $\omega $ - $\lim f_n(y_n) = y$ . So $g_\omega \circ F^{-1} y = \omega $ - $\lim g_n y_n$ by definition of $g_\omega $ . Finally, $F\circ g_\omega \circ F^{-1} = \iota ^{-1}(\omega $ - $\lim f_n(g_ny_n))$ . Now we consider the isometries $\varphi _n(g_n)\in \Gamma $ : they are defined for sufficiently large n since $g_\omega $ displaces $x_\omega $ of some finite quantity. We define the isometry $\iota _*^{-1}(\omega $ - $\lim \varphi _n(g_n))$ : it is an isometry of X, which is proper, and it belongs to $\Gamma $ by Lemma 3.9. We have

$$ \begin{align*}d(\iota_*^{-1}\omega\text{-}\!\lim \varphi_n(g_n)(y), F\circ g_\omega \circ F^{-1}(y)) = \omega\text{-}\!\lim d(\varphi_ng_n(y), f_n(g_ny_n))\end{align*} $$

for every $y\in X$ . Moreover,

$$ \begin{align*} \begin{aligned} d(\varphi_ng_n(y), f_n(g_ny_n)) &\leq d(\varphi_ng_n(y), \varphi_ng_n(f_n(y_n))) + d(\varphi_ng_n(f_n(y_n)), f_n(g_ny_n)) \\ &\leq 2\varepsilon \end{aligned} \end{align*} $$

if n is sufficiently large. This means that $F_* g_\omega = \iota ^{-1} (\omega $ - $\lim \varphi _n g_n) \in \Gamma $ , so $F_*\Gamma _\omega \subseteq \Gamma $ . Now we take $g\in \Gamma $ and we consider the isometries $\psi _n g \in \Gamma _n$ that are defined for sufficiently large n. The sequence $(\psi _n g) $ is admissible and therefore it defines a limit isometry  $g_\omega $ . For all $y\in X$ we have $F_*(g_\omega )(y) = Fg_\omega (y_\omega )$ , where $y_\omega = \omega $ - $\lim y_n$ and $y_n$ is a sequence such that $\iota ^{-1}(\omega $ - $\lim f_n(y_n)) = y$ . Then $F_*(g_\omega )(y) = \iota ^{-1}(\omega $ - $\lim f_n(\psi _n(g)y_n))$ . Once again $\omega $ -a.s. we have

$$ \begin{align*} \begin{aligned} d(F_*(g_\omega)(y), gy) &\leq d(\iota^{-1}(\omega\text{-}\!\lim f_n(\psi_n(g)y_n)), gf_n(y_n)) +\varepsilon\\ &\leq d(f_n(\psi_n(g)y_n), gf_n(y_n)) + 2\varepsilon. \end{aligned} \end{align*} $$

We conclude that $F_*g_\omega = g$ , so $F_*\Gamma _\omega = \Gamma $ .

Proof. We fix $\varepsilon> 0$ . Since $X_\omega $ is proper we can select an ${\varepsilon }/{7}$ -net $S^\varepsilon = \lbrace x_\omega = y_\omega ^1,\ldots ,y_\omega ^{N_\varepsilon }\rbrace $ of $B(x_\omega , {1}/{\varepsilon })$ , where $y_\omega ^i = \omega $ - $\lim y_n^i$ for $i=1,\ldots ,N_\varepsilon $ . Moreover, $\Gamma _\omega $ is closed by Proposition 3.8, so $\Gamma _{\omega ,{1}/{\varepsilon }}(x_\omega )$ is relatively compact by the Arzelà–Ascoli theorem [Reference KelleyKel17, Ch. 7 and Theorem 17]. Therefore we can find a finite subset $g_\omega ^1,\ldots ,g_\omega ^{K_\varepsilon } \in \Gamma _{\omega , {1}/{\varepsilon }}(x_\omega )$ , $g_\omega ^i = \omega $ - $\lim g_n^i$ , with the property that for every $g_\omega \in \Gamma _{\omega , {1}/{\varepsilon }}(x_\omega )$ there exists $1\leq i \leq K_\varepsilon $ such that $d(g_\omega y_\omega , g_\omega ^i y_\omega ) \leq {\varepsilon }/{7}$ for all $y_\omega \in B(x_\omega , {1}/{\varepsilon })$ .

Now $\omega $ -a.s. the following finite set of conditions hold:

• • $\vert d(y_\omega ^i, y_\omega ^j) - d(y_n^i, y_n^j) \vert \leq {\varepsilon }/{7}$ for all $i,j\in \lbrace 1,\ldots , N_\varepsilon \rbrace $ ;

• • the set $S_n^\varepsilon = \lbrace y_n^1,\ldots , y_n^{N_\varepsilon }\rbrace $ is a $\tfrac 27\varepsilon $ -net of $B(x_n,{1}/{\varepsilon })$ ;

• • $g_n^i \in \Gamma _{n,{1}/{\varepsilon }}(x_n)$ for every $i=1,\ldots ,K_\varepsilon $ ;

• • $\vert d(g_\omega ^iy_\omega ^j, y_\omega ^l) - d(g_ny_n^j, y_n^l) \vert \leq {\varepsilon }/{7}$ for all $1\leq i \leq K_\varepsilon $ and all $1\leq j,l \leq N_\varepsilon $ .

• • the set $\lbrace g_n^1,\ldots , g_n^{K_\varepsilon }\rbrace $ is a $\tfrac 27\varepsilon $ -dense subset of $\Gamma _{n,{1}/{\varepsilon }}(x_n)$ with respect to the uniform distance.

For the natural numbers n where these conditions hold we define

$$ \begin{align*}f_n\colon B\bigg(x_n,\frac{1}{\varepsilon}\bigg) \to B\bigg(x_\omega, \frac{1}{\varepsilon}\bigg)\end{align*} $$

by sending the point $y_n$ to a point $y_\omega ^i$ where i is such that $d(y_n,y_n^i) \leq \tfrac 27\varepsilon $ . For $y_n,z_n\in B(x_n,{1}/{\varepsilon })$ we have

$$ \begin{align*}\vert d(f_n(y_n), f_n(z_n)) - d(y_n,z_n)\vert = \vert d(y_\omega^{i_1}, y_\omega^{i_2}) - d(y_n,z_n)\vert\end{align*} $$

for some $i_1,i_2$ . But for these indices n we have $\vert d(y_\omega ^{i_1}, y_\omega ^{i_2}) - d(y_n^{i_1}, y_n^{i_2})\vert \leq \tfrac 27\varepsilon $ , so we get

$$ \begin{align*}\vert d(f_n(y_n), f_n(z_n)) - d(y_n,z_n)\vert \leq \tfrac{6}{7}\varepsilon.\end{align*} $$

Moreover, we define

$$ \begin{align*}\psi_n \colon \Gamma_{\omega,{1}/{\varepsilon}}(x_\omega) \to \Gamma_{n,{1}/{\varepsilon}}(x_n)\end{align*} $$

by sending $g_\omega $ to $g_n^i$ , where $i \in \lbrace 1,\ldots , K_\varepsilon \rbrace $ is such that $d^\infty _{B(x_\omega , 1/\varepsilon )}(g_\omega , g_\omega ^i) \leq {\varepsilon }/{7}$ .

Let $g_\omega \in \Gamma _{\omega ,{1}/{\varepsilon }}(x_\omega )$ , so $\Psi _n(g_\omega )=g_n^i$ as before. Let $y_n \in B(x_n,{1}/{\varepsilon })$ such that also $g_n^iy_n\in B(x_n,{1}/{\varepsilon })$ . Let $j,l\in \lbrace 1,\ldots , N_\varepsilon \rbrace $ be such that $d(y_n,y_n^j) \leq \tfrac 27\varepsilon $ and $d(g_n^iy_n, y_n^l)\leq \tfrac 27\varepsilon $ . By definition $f_n(y_n)=y_\omega ^i$ , while $f_n(\psi _n(g_\omega )y_n) = y_\omega ^l$ . We have

$$ \begin{align*} \begin{aligned} d(f_n(\psi_n(g_\omega)y_n), g_\omega f_n(y_n)) &= d(y_\omega^l, g_\omega y_\omega^j)\\ &\leq d(y_\omega^l, g_\omega^i y_\omega^j) + \tfrac{2}{7}\varepsilon \\ &\leq d(y_n^l, g_n^i y_n^j) + \tfrac{3}{7}\varepsilon \leq \varepsilon. \end{aligned} \end{align*} $$

Finally, we define

$$ \begin{align*}\varphi_n \colon \Gamma_{n,{1}/{\varepsilon}}(x_n) \to \Gamma_{\omega, {1}/{\varepsilon}}(x_\omega)\end{align*} $$

as $\varphi _n(g_n)=g_\omega ^i$ , where $d^\infty _{B(x_n,1/\varepsilon )}(g_n, g_n^i)\leq \tfrac 27\varepsilon $ .

Let $g_n \in \Gamma _{n,{1}/{\varepsilon }}(x_n)$ and let $\varphi _n(g_n)=g_\omega ^i$ . Now let $y_n\in B(x_n,{1}/{\varepsilon })$ such that also $g_ny_n \in B(x_n, {1}/{\varepsilon })$ . Let $j,l\in \lbrace 1,\ldots , N_\varepsilon \rbrace $ such that $d(y_n, y_n^j)\leq \tfrac 27\varepsilon $ and $d(g_ny_n, y_n^l)\leq \tfrac 27\varepsilon $ , so that $f_n(y_n)=y_\omega ^j$ and $f_n(g_ny_n)=y_\omega ^l$ . Therefore

$$ \begin{align*} \begin{aligned} d(f_n(g_ny_n), \varphi_n(g_n)f_n(y_n)) &= d(y_\omega^l, g_\omega^i y_\omega^j)\\ &\leq d(y_n^l, g_n y_n^j) + \tfrac{3}{7}\varepsilon \\ &\leq d(y_n^l, g_n y_n) + \tfrac{5}{7}\varepsilon \leq \varepsilon. \end{aligned} \end{align*} $$

This shows that $\omega $ -a.s. we can find an equivariant $\varepsilon $ -approximation between $(X_n,x_n,\Gamma _n)$ and $(X_\omega , x_\omega , \Gamma _\omega )$ . For all integers k we set $\varepsilon = {1}/{k}$ and we choose $n_k \in \mathbb {N}$ in the set of indices for which there exists an equivariant ${1}/{k}$ -approximation as before. The sequence $(X_{n_k}, x_{n_k}, \Gamma _{n_k})$ satisfies the thesis.

Proof. The equivariant pointed Gromov–Hausdorff convergence is metrizable (cf. [Reference FukayaFuk86]), so it is enough to show that every subsequence has a subsequence that converges to $(X,x,\Gamma )$ . Fix a subsequence $\lbrace n_k \rbrace $ . The set $\{n_k\}$ is infinite; then there exists a non-principal ultrafilter $\omega $ containing it for which $\omega (\{n_k\})=1$ (cf. [Reference JansenJan17, Lemma 3.2]). The ultralimit with respect to $\omega $ of the sequence $(X_{n_k},x_{n_k},\Gamma _{n_k})$ is the same as that of the sequence $(X_n,x_n,\Gamma _n)$ since $\omega (\{n_k\})=1$ . By Proposition 3.13 we can extract a further subsequence $\lbrace n_{k_j} \rbrace $ that converges in the equivariant pointed Gromov–Hausdorff sense to $(X,x,\Gamma )$ .

Proof. If $z'\in B(z,\rho )$ then $(z,z')_{x}> \log ({1}/{\rho })$ , so $D_{x,a}(z,z')\leq Ve^{-a(z,z')_{x}} < V\rho ^a.$ If $z'\in B_{D_{x,a}}(z,({1}/{V})\rho ^a)$ then $({1}/{V})e^{-a(z,z')_{x}} \leq D_{x_0,a}(z,z') < ({1}/{V})\rho ^a$ , that is, $z'\in B(z,\rho )$ .

Proof. Let $z'\in B(z,e^{-T})$ , that is, $(z,z')_{x}> T$ . By Lemma 4.2 we know that $d(\xi _{x,z}(T - \delta ), \xi _{x,z'}(T - \delta )) \leq 4\delta .$ So $d(\xi _{x,z'}(T), \xi _{x,z}(T)) \leq 6\delta < 7\delta $ . This implies $z'\in \text {Shad}_{x}(\xi _{x,z}(T),7\delta )$ , showing (i).

Now we fix $z'\in \text {Shad}_{x}(\xi _{x,z}(T),r)$ , which means that every geodesic ray $\xi _{x,z'}$ passes through $B(\xi _{x,z}(T), r)$ , so $d(\xi _{x,z'}(T),\xi _{x,z}(T)) < 2r$ . By Lemma 4.2 we conclude $(z,z')_{x}> T - r$ , implying (ii).

Proof. Let $z_1, \ldots , z_N$ be points of C realizing $\mathrm {Cov}(C, e^{-T})$ . Suppose there exist points $w_1,\ldots ,w_M$ of C such that $B(w_i, e^{-T +\delta })$ are disjoint, in particular $(w_i,w_j)_{x}\leq T - \delta $ for every $i\neq j$ . If $M> N$ then two different points $w_i,w_j$ belong to the same ball $B(z_k, e^{-T})$ , that is, $(z_k,w_i)_{x}> T$ and $(z_k,w_j)_{x}> T.$ By (4.3) we have $(w_i,w_j)_{x}> T - \delta $ , which is a contradiction. This shows the first inequality.

Now let $z_1,\ldots ,z_N$ be a maximal collection of points of C such that the $B(z_i, e^{-T})$ are disjoint. Then for every $z\in C$ there exists i such that $B(z,e^{-T}) \cap B(z_i,e^{-T}) \neq \emptyset $ . Therefore there exists $w\in \partial X$ such that $(z_i,w)_{x}> T$ and $(z,w)_{x}> T$ . As before, we get $(z_i,z)_{x}> T- \delta ,$ proving the second inequality.

Proof. The proof is classical for geodesic metric spaces. In this setting we need to use the fact that $\textrm{QC-Hull}(\Lambda (\Gamma ))$ is $36\delta $ -quasiconvex. By discreteness we can fix a small $\varepsilon>0$ such that $d(x,gx) < 2D+72\delta + \varepsilon $ implies $d(x,gx)\leq 2D+72\delta $ for all $g\in \Gamma $ . We take any $g\in \Gamma $ and we take consecutive points $x_i$ , $i=0,\ldots ,N$ , on a geodesic segment $[x,gx]$ such that $x_0=x$ , $x_N=gx$ and $d(x_i,x_{i+1}) <\varepsilon $ . By Lemma 4.5 each $x_i$ is at distance at most $36\delta $ from a point $y_i \in \textrm{QC-Hull}(\Lambda (\Gamma ))$ , hence there exists some $h_i\in \Gamma $ such that $d(h_ix,x_i)\leq 36\delta + D$ . We can choose $h_N = g$ and $h_0=\text {id}$ . We define the elements $g_i = h_{i-1}^{-1} h_i$ for $i = 1,\ldots ,N$ . Clearly $g_1\cdots g_{N-1}g_N = g$ . Moreover, $d(g_i x, x) < 72\delta + 2D + \varepsilon $ for every i, so $g_i \in \Sigma _{2D+72\delta }(x)$ .

Proof. Using the same proof as for [Reference Besson, Courtois, Gallot and SambusettiBCGS17, Lemma 5.14], we conclude that there exists a hyperbolic isometry $a\in \Gamma $ such that $\ell (a)\leq 8D + 10\delta $ . The remaining part of the proof can be done exactly in the same way as in [Reference Besson, Courtois, Gallot and SambusettiBCGS17], choosing $y\in \text {Min}(a) \subseteq \textrm{QC-Hull}(\Lambda (\Gamma ))$ and using Lemma 5.1.

Proof. The proof is the same as for [Reference Besson, Courtois, Gallot and SambusettiBCGS21, Theorem 3.4]. The only non-trivial part is the Bishop–Gromov estimate stated in [Reference Besson, Courtois, Gallot and SambusettiBCGS21, Theorem 3.1] and proved in [Reference Besson, Courtois, Gallot and SambusettiBCGS17, Theorem 5.1]. It is made in the cocompact case but it extends word for word to the quasiconvex-cocompact setting.

Proof. We prove only that Y is uniformly covered since the other case is similar. We fix $0<r\leq R$ . The map $y\mapsto \text {Cov}(\overline {B}(y,R) \cap Y,r)$ defined on Y is clearly $\Gamma $ -invariant. If the thesis is false we could find a sequence of points $x_n\in Y$ such that $\text {Cov}(\overline {B}(x_n,R) \cap Y,r) \geq n$ . By $\Gamma $ -invariance and the compactness of the quotient we can suppose that $x_n$ converges to some point $x_\infty $ that belongs to Y by Lemma 4.5. Clearly we would have $\text {Cov}(\overline {B}(x_\infty ,R+1) \cap Y,r) = \infty $ , which is impossible since $\overline {B}(x_\infty \cap Y,R+1)$ is compact.

Proof. The proof is the same as for [Reference Cavallucci and SambusettiCS21, Lemma 4.7], except for the fact that Y is not geodesic but only $36\delta $ -quasigeodesic by Lemma 4.5. We proceed by induction on k, where k is the smallest integer such that $T\leq 72\delta + 3r + kr$ . For $k=0$ the result is obvious by our assumption. The inductive step goes as follows: by induction we can find a maximal $2r$ -separated subset $\lbrace y_1,\ldots ,y_N\rbrace $ of $\overline {B}(x,T-r)\cap Y$ with $N\leq P(1+P)^{(({T-r})/{r})-1}$ . The key step is to show that $\bigcup _{i=1}^N \overline {B}(y_i,72\delta +3r) \supseteq A(x,T-r,T)\cap Y$ , where $A(x,T-r,T)$ is the closed annulus centred at x of radii $T-r$ and T. Indeed, for every point $y\in A(x,T-r,T)$ we consider the point $y'$ along a geodesic segment $[x,y]$ at distance $T - r - 36\delta $ from x. By quasiconvexity there is a point $z\in Y$ at distance no greater than $36\delta $ from $y'$ . In particular, $z\in \overline {B}(x,T-r)$ , so $d(z,y_i)\leq 2r$ for some $i=1,\ldots ,N$ . We conclude that $d(y,y_i)\leq 72\delta + 3r$ . The rest of the proof can be done exactly as for [Reference Cavallucci and SambusettiCS21, Lemma 4.7], while the estimate on the upper covering entropy follows trivially using (5.1).

Proof. Let us fix $0<r\leq r'$ . We have

$$ \begin{align*}\mathrm{Cov}(\overline{B}(x,T)\cap Y,r') \leq \mathrm{Cov}(\overline{B}(x,T)\cap Y,r)\end{align*} $$

and

$$ \begin{align*}\mathrm{Cov}(\overline{B}(x,T)\cap Y,r) \leq \mathrm{Cov}(\overline{B}(x,T)\cap Y,r') \cdot \text{Cov}_Y(r',r).\end{align*} $$

The quantity $\text {Cov}_Y(r',r)$ is finite by Lemma 5.4. These inequalities easily imply that $\overline {h}_{\mathrm {Cov}}(Y,r) = \overline {h}_{\mathrm {Cov}}(Y,r')$ and $\underline {h}_{\mathrm {Cov}}(Y,r) = \underline {h}_{\mathrm {Cov}}(Y,r')$ . Moreover, by (5.1) these quantities can be defined by replacing the covering function with the packing function. Furthermore, an application of Lemmas 5.4 and 5.6 shows that the upper covering entropy of Y is finite.

Let $2s=\text {sys}(\Gamma ,X)>0$ . We have

$$ \begin{align*}\text{Cov}(\overline{B}(x,T)\cap Y,D) \leq \#\Gamma x \cap \overline{B}(x,T+D)\end{align*} $$

and

$$ \begin{align*}\text{Pack}(\overline{B}(x,T)\cap Y,s) \geq \#\Gamma x \cap \overline{B}(x,T).\end{align*} $$

Observe that the sequence $\#\Gamma x \cap \overline {B}(x,T)$ converges to $h_\Gamma $ when T goes to $+\infty $ by Lemma 4.12. Therefore $\overline {h}_{\mathrm {Cov}}(Y,D) \leq h_\Gamma $ and $\underline {h}_{\mathrm {Cov}}(Y,s) \geq h_\Gamma $ , implying the last part of the thesis.

Proof. By the Gromov precompactness theorem [Reference GromovGro81] we know that

$$ \begin{align*}\sup_{n\in\mathbb{N}}\mathrm{Pack}(\overline{B}(x_n,R+D)\cap Y_n,r) =: P < +\infty.\end{align*} $$

For every $n\in \mathbb {N}$ and every point $y_n \in Y_n$ there is some $g\in \Gamma _n$ such that $d(y_n,g_nx_n)\leq D$ . Therefore

$$ \begin{align*} \begin{aligned} \mathrm{Pack}(\overline{B}(y_n,R) \cap Y_n,r) &\leq \mathrm{Pack}(\overline{B}(gx_n,R+D)\cap Y_n,r) \\ &= \mathrm{Pack}(\overline{B}(x_n,R+D)\cap Y_n,r) \leq P \end{aligned} \end{align*} $$

by the $\Gamma _n$ -invariance of $Y_n$ .

Proof. By Lemma 5.8 we have $\sup _{n\in \mathbb {N}}\text {Pack}_{Y_n}(72\delta + 3,1) =: P < +\infty $ . Therefore by Proposition 5.7 and Lemma 5.6 we have

$$ \begin{align*}h_{\Gamma_n} = h_{\mathrm{Cov}}(Y_n) \leq \log(1+P).\\[-3pc]\end{align*} $$

Proof. By Corollary 5.9 and Proposition 5.3 there is some $s>0$ such that $\text {sys}(\Gamma _n,X_n) \geq s$ for every $n\in \mathbb {N}$ . Let $\omega $ be a non-principal ultrafilter. By Proposition 3.13 it is enough to show that $\Gamma _\omega $ is discrete and torsion-free. Let $g_\omega = \omega $ - $\lim g_n$ be a non-trivial element of $\Gamma _\omega $ and $y_\omega = \omega $ - $\lim y_n$ be a point of $X_\omega $ . We know that $d(y_n, g_n y_n )\geq s$ for $\omega $ -a.e. $(n)$ . This implies $d(y_\omega , g_\omega y_\omega )\geq s$ . Since this is true for every $g_\omega \in \Gamma _\omega $ and every $y_\omega \in X_\omega $ we conclude that sys $(\Gamma _\omega , X_\omega ) \geq s$ . Since $X_\omega $ is proper we conclude that $\Gamma _\omega $ is discrete. Now take an elliptic element $g_\omega =\omega $ - $\lim g_n$ . It is classical that $g_\omega $ must have finite order since $\Gamma _\omega $ is discrete (see, for instance, [Reference Besson, Courtois, Gallot and SambusettiBCGS17, Remark 8.16]), that is, $g_\omega ^k =\text {id}$ for some $k\in \mathbb {Z}\setminus \lbrace 0 \rbrace $ . This means that $\omega $ - $\lim d(g_n^k x_n, x_n) = 0$ , so $g_n^k = \text {id}$ for $\omega $ -a.e. $(n)$ . This implies $g_n = \text {id}$ for $\omega $ -a.e. $(n)$ and therefore $g_\omega = \text {id}$ . In other words, $\Gamma _\omega $ is torsion-free.

Proof. A point of $\omega $ - $\lim \partial X_n$ is a class of a sequence of points $(z_n) \in \partial X_n$ and for each point $z_n$ there exists a geodesic ray $\xi _{x_n,z_n}$ . The sequence of geodesic rays $(\xi _{x_n,z_n})$ defines an ultralimit geodesic ray $\xi $ of $X_\omega $ with $\xi (0)=x_\omega $ (cf. [Reference Cavallucci and SambusettiCS21, Lemma A.7]) which provides a point of $\partial X_\omega $ . We denote this point by $z_\omega $ and $\xi $ by $\xi _{x_\omega ,z_\omega }$ . We define the map $\Psi \colon \omega $ - $\lim \partial X_n \to \partial X_\omega $ as $\Psi ((z_n)) = \xi _{x_\omega ,z_\omega }^+ = z_\omega $ .

Good definition. We need to show that $\Psi $ is well defined, that is, it does not depend on the choice of the geodesic ray $\xi _{x_n,z_n}$ or on the choice of the sequence $(z_n)$ . Let $(z_n')$ be a sequence of points equivalent to $(z_n)$ , that is, $\omega $ - $\lim D_{x_n,a}(z_n,z_n') = 0$ . Choose geodesic rays $\xi _{x_n,z_n}$ , $\xi _{x_n,z_n'}$ . For every n the metric $D_{x_n,a}$ is a standard visual metric. Then for every fixed $\varepsilon> 0$ we have $(z_n,z_n')_{x_n}> \log ({1}/{\varepsilon }) =: T_\varepsilon $ for $\omega $ -a.e. $(n)$ . Thus $d(\xi _{x_n,z_n}(T_\varepsilon - \delta ), \xi _{x_n,z_n'}(T_\varepsilon - \delta )) \leq 4\delta $ by Lemma 4.2, $\omega $ -a.s. We conclude that $d(\xi _{x_\omega ,z_\omega }(T_\varepsilon - \delta ), \xi _{x_\omega ,z_\omega '}(T_\varepsilon - \delta )) < 6\delta \omega $ -a.s. Therefore, again by Lemma 4.2, $(z_\omega ,z_\omega ')_{x_\omega }> T_\varepsilon - 4\delta $ . Thus $(z_\omega ,z_\omega ')_{x_\omega } = +\infty $ by the arbitrariness of $\varepsilon $ , that is, $z_\omega = z_\omega '$ .

Injectivity. The next step is to show that $\Psi $ is injective. If two sequences of points $(z_n), (z_n')$ have the same image under $\Psi $ then $(\xi _{x_\omega ,z_\omega }^+, \xi _{x_\omega ,z_\omega '}^+)_{x_\omega } = +\infty $ . So $(\xi _{x_\omega ,z_\omega }^+, \xi _{x_\omega ,z_\omega '}^+)_{x_\omega } \geq T$ for every fixed $T\geq 0$ . Hence $d(\xi _{x_\omega ,z_\omega }(T - \delta ), \xi _{x_\omega ,z_\omega '}(T - \delta )) \leq 4\delta $ , by Lemma 4.2. Then $d(\xi _{x_n,z_n}(T - \delta ), \xi _{x_n,z_n'}(T - \delta )) < 6\delta \omega $ -a.s., that is, $(z_n,z_n')_{x_n}> T- 4\delta \omega $ -a.s., again by Lemma 4.2. Therefore $D_{x_n,a}(z_n,z_n')\leq e^{-a(T - 4\delta )}$ . Since this is true $\omega $ -a.s. we get $\omega $ - $\lim D_{x_n,a}(z_n,z_n') \leq e^{-a(T - 4\delta )}$ for $\omega $ -a.e. $(n)$ . By the arbitrariness of T we deduce that $\omega $ - $\lim D_{x_n,a}(z_n,z_n') = 0$ , that is, $(z_n) = (z_n')$ as elements of $\omega $ - $\lim \partial X_n$ .

Homeomorphism. Let us show $\Psi $ is continuous. As both $\omega $ - $\lim \partial X_n$ and $\partial X_\omega $ are metrizable, it is enough to check the continuity on sequences of points. We take a sequence $(z_n^k)_{k\in \mathbb {N}}$ converging to $(z_n^\infty )$ in $\omega $ - $\lim \partial X_n$ . By definition for every $\varepsilon> 0$ there exists $k_\varepsilon \geq 0$ such that if $k\geq k_\varepsilon $ then $\omega $ - $\lim D_{x_n,a}(z_n^k, z_n^\infty ) <\varepsilon $ . Therefore for every fixed $k\geq k_\varepsilon $ we have $(z_n^k,z_n^\infty )_{x_n} \geq \log ({1}/{\varepsilon }) =: T_\varepsilon $ for $\omega $ -a.e. $(n)$ . As usual, we conclude that $d(\xi _{x_nz_n^k}(T_\varepsilon - \delta ),\xi _{x_nz_n^\infty }(T_\varepsilon - \delta )) \leq 4\delta \omega $ -a.s. Thus $d(\xi _{x_\omega z_\omega ^k}(T_\varepsilon - \delta ),\xi _{x_\omega ,z_\omega ^\infty }(T_\varepsilon - \delta )) < 6\delta $ for every fixed $k\geq k_\varepsilon $ . Again this implies $(z_\omega ^k,z_\omega ^\infty )_{x_\omega }> T_\varepsilon - 4\delta $ for all $k\geq k_\varepsilon $ . By the arbitrariness of $\varepsilon $ we get that $z_\omega ^k$ converges to $z_\omega ^\infty $ when k goes to $+\infty $ .

The continuity of the inverse map defined on the image of $\Psi $ can be proved in a similar way.

Proof of Theorem A(i)

In order to simplify the notation we fix a non-principal ultrafilter  $\omega $ . We know that $(X_\omega , x_\omega , \Gamma _\omega )$ is equivariantly isometric to $(X_\infty , x_\infty , \Gamma _\infty )$ by Proposition 3.13 and that $X_\omega $ is a proper metric space, so we can prove all the properties for this triple. It is classical that the ultralimit of geodesic, $\delta $ -hyperbolic metric spaces is a geodesic and $\delta $ -hyperbolic metric space; see, for instance, [Reference Druţu and KapovichDK18]. Moreover, by Corollary 5.10 the group $\Gamma _\omega $ is discrete and torsion-free.

Let $\Psi $ be the homeomorphism onto the image given by Proposition 5.11. We claim that $\Lambda (\Gamma _\omega ) = \Psi (\omega $ - $\lim \Lambda (\Gamma _n))$ . We fix a sequence $z_n \in \Lambda (\Gamma _n)$ and we observe that by Lemma 4.6 and the cocompactness of the action of $\Gamma _n$ on $\textrm{QC-Hull}(\Lambda (\Gamma _n))$ we can find a sequence $(g_n^k)_{k\in \mathbb {N}}\subseteq \Gamma _n$ such that, denoting one geodesic ray $[x_n,z_n]$ by $\xi _{x_n,z_n}$ , we have:

1. (a) $g_n^k x_n$ converges to $z_n$ when k tends to $+\infty $ ;

2. (b) $g_n^0 = \text {id}$ ;

3. (c) $d(g_n^k x_n, g_n^{k+1}x_n)\leq 28\delta + 2D$ ;

4. (d) $d(g_n^k x_n, \xi _{x_n,z_n}(k)) \leq 14\delta + D$ .

For every $k \in \mathbb {N}$ the sequence $g_n^k$ is admissible by (b) and (c), so it defines a limit isometry $g_\omega ^k \in \Gamma _\omega $ . Moreover, if $\xi _{x_\omega ,z_\omega }$ is the ultralimit of the sequence of geodesic rays $\xi _{x_n,z_n}$ , we have $d(g_\omega ^k x_\omega , \xi _{x_\omega ,z_\omega }(k)) \leq 14\delta + D$ for every $k\in \mathbb {N}$ . Observe that $\xi _{x_\omega ,z_\omega }^+ = \Psi ((z_n))$ by definition of $\Psi $ . As a consequence the sequence $g_\omega ^k x_\omega $ converges to $\Psi ((z_n))$ , that is, $\Psi (z_n)\in \Lambda (\Gamma _\omega )$ . This shows that $\Psi (\omega $ - $\lim \Lambda (\Gamma _n)) \subseteq \Lambda (\Gamma _\omega )$ . Clearly $\Gamma _\omega $ acts on $\omega $ - $\lim \Lambda (\Gamma _n)$ by $(g_n)(z_n) = (g_n z_n)$ and this action commutes with $\Psi $ . The set $\omega $ - $\lim \Lambda (\Gamma _n)$ is $\Gamma _\omega $ -invariant and closed. The $\Gamma _\omega $ -invariance is trivial, so let us check the closure. If $(z_n^k)_{k\in \mathbb {N}} \in \omega $ - $\lim \Lambda (\Gamma _n)$ is a sequence converging to $(z_n^\infty )$ and $z_n^\infty \notin \omega $ - $\lim \Lambda (\Gamma _n)$ then there exists $\varepsilon> 0$ such that $D_{x_n,a}(z_n^\infty , \Lambda (\Gamma _n)) \geq \varepsilon \omega $ -a.s. This is a contradiction. Therefore the set $\Psi (\omega $ - $\lim \Lambda (\Gamma _n))$ is also closed and $\Gamma _\omega $ -invariant. By Proposition 4.11 we conclude that $\Lambda (\Gamma _\omega ) = \Psi (\omega $ - $\lim \Lambda (\Gamma _n))$ . This also implies that $\omega $ - $\lim \textrm{QC-Hull}(\Lambda (\Gamma _n)) = \textrm{QC-Hull}(\Lambda (\Gamma _\omega ))$ and so $x_\omega \in \textrm{QC-Hull}(\Lambda (\Gamma _\omega ))$ . For every pair of points $y_\omega , y_\omega ' \in \textrm{QC-Hull}(\Lambda (\Gamma _\omega ))$ there exist sequences of points $y_n, y_n' \in \textrm{QC-Hull}(\Lambda (\Gamma _n))$ such that $y_\omega = \omega $ - $\lim y_n$ and $y_\omega ' = \omega $ - $\lim y_n'$ . So there exists $g_n \in \Gamma _n$ such that $d(g_ny_n,y_n')\leq D$ . The sequence $g_n$ is clearly admissible so it defines an element $g_\omega = \omega $ - $\lim g_n$ of $\Gamma _\omega $ and $d(g_\omega y_\omega , y_\omega ')\leq D$ , implying that the action of $\Gamma _\omega $ on $\textrm{QC-Hull}(\Lambda (\Gamma _\omega ))$ is cocompact with codiameter less than or equal to D. It remains only to show that $\Gamma _\omega $ is non-elementary. If $\Gamma _\omega $ is elementary then $\textrm{QC-Hull}(\Lambda (\Gamma _\omega )) = \mathbb {R}$ and $\Gamma _\omega $ acts on $\mathbb {R}$ as $\mathbb {Z}_\tau $ , the group generated by the translation of length $\tau $ , for some $\tau> 0$ . Denote by $g_\omega = \omega $ - $\lim g_n$ the element corresponding to this translation. For every $k\in \mathbb {N}$ we notice that

$$ \begin{align*}A_\omega(k)=\left\lbrace h_\omega \in \Gamma_\omega \text{ s.t. } d(x_\omega, h_\omega x_\omega) < (k+1)\cdot\tau\right\rbrace = \lbrace g_\omega ^{\pm m} \rbrace:{m=0,\ldots, k}.\end{align*} $$

In particular, $A_\omega (k)$ has cardinality $2k + 1$ . We define also the sets

$$ \begin{align*}A_n(k)=\big\lbrace h_n \in \Gamma_n \text{ s.t. } d(x_n, h_n x_n) \leq \big(k+\tfrac{1}{2}\big)\cdot\tau\big\rbrace.\end{align*} $$

Since we have a uniform bound on the systole and the action is torsion-free, we have that $\#A_n(k) \leq \#A_\omega (k) \omega $ -a.s., for every fixed $k\in \mathbb {N}$ . We apply this property to $k_0=({72\delta + 2D})/{\tau }$ . Clearly $g_n^{\pm m} \in A_n(k_0)$ for every $m=0,\ldots ,k_0$ , $\omega $ -a.s. Therefore $A_n(k_0)=\lbrace g_n^{\pm m} \rbrace _{m=0,\ldots ,k_0} \omega $ -a.s. By Lemma 5.1 we conclude that

$$ \begin{align*}\Gamma_n = \langle A_n(k_0) \rangle = \langle g_n \rangle,\end{align*} $$

that is, $\Gamma _n$ is elementary $\omega $ -a.s., which is a contradiction.