Proof of Theorem 2.7

The fact that $B_-$ and $B_+$ are amenable Lebesgue G-spaces such that the diagonal G-action on $B_-\times B_+$ is metrically ergodic is proven in [Reference Bader and Furman3, Theorem 2.7]. We will show that diagonal G-action on $B_-\times B_+$ is coarsely metrically ergodic.

Fix a G-invariant Borel metric $d:(B_-\times B_+)^2\to [0,\infty )$ . By Remark 2.6 and upon normalization, we assume, as we may, that d is $1$ -separable. We will show that for a.e. $b_-\in B_-$ , the essential diameter of $\{b_-\}\times B_+$ is bounded by $3$ . By symmetry, for a.e. $b_+\in B_+$ , the essential diameter of $B_-\times \{b_+\}$ is bounded by $3$ , and thus the essential diameter of $B_-\times B_+$ is bounded by $6$ .

We find a countable collection $A_i$ of non-null $1$ -balls whose union is of full measure. We consider the projection $B_-\times B_+\to B_-$ and for every i, we let $A_i^-$ be the image of $A_i$ . For every natural n, we let $A_{i,n}^-$ be the set of those $b_-\in A_i^-$ such that

$$ \begin{align*} \nu_+(\{b_+\in B_+\mid (b_-,b_+)\in A_i\}) \geq 1/n. \end{align*} $$

Note that there exists $N_i$ such that the sets $A_{i,n}^-$ are non-null for every $n\geq N_i$ . Using the symbol $\circeq $ for equation up to null-sets, we get $A_{i}^-\circeq \bigcup _{n\geq N_i}~ A_{i,n}^-$ . We let $A_{i,n}$ be the preimage of $A_{i,n}^-$ in $A_i$ and get

$$ \begin{align*} A_{i}\circeq\bigcup_{n\geq N_i}~ A_{i,n} \quad \mbox{and} \quad B_+\times B_-\circeq \bigcup_i~ A_{i}\circeq\bigcup_{i,{n\geq N_i}}~ A_{i,n}. \end{align*} $$

By modifying the sets $A_{i,n}$ , we assume as we may that for every $i',n',i",n"$ , $A_{i',n'}^-\cap A_{i",n"}^-$ is either non-null or empty, upon throwing away the preimages of the intersection in case it is null. By Fubini, for a.e. $b_-\in B_-$ , for a.e. $b_+\in B_+$ , $(b_-,b_+)\in \bigcup _{i,{n\geq N_i}}~ A_{i,n}$ . We fix a generic $b_-\in B_-$ and consider the full measure subset of the fibre over $b_-$ ,

$$ \begin{align*} D=(\{b_-\}\times B_+)\cap \bigg(\bigcup_{i,{n\geq N_i}}~ A_{i,n}\bigg). \end{align*} $$

We claim that the set D is bounded by $3$ . To see this, let us fix $b_{+}^{\prime },b_+^{\prime \prime }\in B_+$ such that $(b_-,b_{+}^{\prime }),(b_-,b_+^{\prime \prime })\in D$ . Fix $i',n',i",n"$ such that $(b_-,b_{+}^{\prime })\in A_{i',n'}$ , $(b_-,b_+^{\prime \prime })\in A_{i",n"}$ and $n'>N_{i'}$ , $n">N_{i"}$ . Set

$$ \begin{align*}\epsilon=\min\{1/n',1/n"\} \quad \text{and} \quad E^-=A_{i',n'}^-\cap A_{i",n"}^-.\end{align*} $$

Let E be the preimage of $E^-$ in $A_{i',n'}$ and note that it is non-null. By Lemma 2.8, there exist $g\in G$ and a non-null subset $E_1^-$ in $E^-\cap g^{-1}E^-$ such that for every $b_{-}^{\prime }\in E_1^-$ ,

$$ \begin{align*} \nu_+(gE_{b_{-}^{\prime}})> 1-\epsilon, \end{align*} $$

where $E_{b_{-}^{\prime }}=\{b_+\mid (b_{-}^{\prime },b_+)\in E\}$ . Fixing such $b_{-}^{\prime }$ , we conclude that the set $gE_{b_{-}^{\prime }}$ is bounded by $1$ and it intersects both sets $A_{i',n'}$ and $A_{i",n"}$ . We conclude that $d((b_-,b_{+}^{\prime }),(b_-,b_+^{\prime \prime }))<3$ . This finishes the proof.

Proof. We fix a maximal compact subgroup $K<G$ . In [Reference Furstenberg10], Furstenberg proved that all bounded harmonic functions on the symmetric space $G/K$ are represented via a Poisson formula as bounded measurable functions on $G/P$ with respect to a certain G-quasi-invariant measure $\nu \in \operatorname {Prob}(G/P)$ , namely the unique K-invariant measure. Following the discretization process applied by Furstenberg in the special case of the hyperbolic plane, Lyons and Sullivan obtained in [Reference Lyons and Sullivan14, Theorem 5] a discretization scheme which applies to any *-recurrent discrete subset of a Riemannian manifold of bounded geometry (see [Reference Lyons and Sullivan14, §7] for the definition of *-recurrent), an example of such a setting being the orbit of $\Gamma $ on $G/K$ . Their discretization procedure for the pair $(G/K,\Gamma .K)$ commutes with the symmetries of the pair, and thus provides a probability measure $\mu $ on $\Gamma $ for which the bounded $\mu $ -harmonic functions are exactly the restriction to $\Gamma $ of the bounded harmonic functions on $G/K$ . That is, the future Furstenberg–Poisson boundary $B_+$ of $(\Gamma ,\mu )$ is $(G/P,\nu )$ . The main theorem of [Reference Ballmann and Ledrappier4] shows further that the measure $\mu $ could be chosen to be symmetric, that is, $\check {\mu }=\mu $ . This shows that the past Furstenberg–Poisson boundary $B_-$ of $(\Gamma ,\mu )$ is $(G/P,\nu )$ as well. We are thus done by Theorem 2.7.

Proof. Given a coarsely invariant coarse metric d on $\Omega $ , observe that $d'(x,y)=\sup \{d(gx,gy)\mid g\in \Gamma \}$ is a $\Gamma $ -invariant measurable coarse metric. Its boundedness implies the boundedness of d. The last part of the lemma follows from the fact that the function $x\mapsto d'(x,x)$ is automatically bounded, by Remark 2.5.

Proof. We choose a countable C-dense subset V in U, identify it with $\mathbb {N}$ and define $\pi $ inductively, sending points at the C-ball around $v\in V$ to v, unless their $\pi $ value was already determined. We set $d'=d|_V$ .

Proof. The first assertions of the lemma are straightforward. For the last assertion under the extra hypothesis that U is countable, we observe that the infimum in the definition of $\bar d$ could be taken over the countable collection of the finite subsets of U. This ensures that $\bar d$ is indeed Borel.

Proof. We assume $f:\Omega \to \operatorname {Prob}(U)$ is a Borel $\Gamma $ -map and d is a coarsely separable $\Gamma $ -invariant Borel coarse metric on U, and we argue to show that the $\Gamma $ -orbits in U are d-bounded. Since the restriction of $\bar {d}$ to U is at a bounded distance away from d (see Lemma 2.14), it is enough to show that $\operatorname {Prob}(U)$ contains a $\bar {d}$ -bounded subset.

We let $\pi :U\to V$ and $d'$ be as in Lemma 2.13, so that the pull back $\pi ^{*}d'$ is coarsely equivalent to d. It follows $\overline {\pi ^{*}d'}$ is coarsely equivalent to $\bar {d}$ . By Lemma 2.14, $\bar {d'}$ is Borel, symmetric and it satisfies the coarse triangle inequality. Hence, the same applies also to $\overline {\pi ^{*}d'}=(\pi _*)^{*}\bar {d'}$ .

By Lemma 2.11, $f^{*}\overline {\pi ^{*}d'}$ is a bounded coarse metric on $\Omega $ . We deduce that $\operatorname {Prob}(U)$ contains a $\overline {\pi ^{*}d'}$ -bounded subset. The proof is now complete, as this subset is $\bar {d}$ -bounded.

Proof. By the fact that $1-\epsilon>1/2$ , all the Borel subsets A with $m(A)\geq 1-\epsilon $ intersect in pairs, so the diameter of $\beta _\epsilon (m)$ is bounded by $2r_\epsilon (m)+2$ . Equivariance is clear.

Let now $C,m_1,m_2$ as in the last statement. Note that every Borel subset A of U with $m_1(A)\geq 1-\epsilon /C$ satisfies $m_2(U\setminus A)\leq Cm_1(U\setminus A) \leq \epsilon $ , and hence $m_2(A)\geq 1-\epsilon $ . It follows that $r_{\epsilon }(m_2)\leq r_{\epsilon /C}(m_1)$ .

If $u\in \beta _{\epsilon }(m_2)$ , then by definition, u belongs to a Borel set A with $m_2(A)\geq 1-\epsilon $ and diameter less than $r_{\epsilon }(m_2)+1\leq r_{\epsilon /C}(m_1)+1$ . Fix any Borel set B of diameter less than $r_{\epsilon /C}(m_1)+1$ and $m_1(B)\geq 1-\epsilon /C$ . Then, $m_2(B)\geq 1-\epsilon $ and we get $A\cap B\neq \emptyset $ . In particular, u lies within $r_{\epsilon /C}(m_1)+1$ of B, whence of $\beta _{\epsilon /C}(m_1)$ .

Proof. The fact that i is continuous is obvious. For $x\neq y$ in X, note that the difference function $d(x,\cdot )-d(y,\cdot )$ is not constant, as it attains different values at x and y. Thus, i is injective. We now fix $x\in X$ . First, we note that $\tilde {X}_x$ is closed subset of

$$ \begin{align*} \prod_{y\in X} [-d(x,y),d(x,y)] \subset \prod_{y\in X} \mathbb{R}=\mathbb{R}^X, \end{align*} $$

and thus it is compact. Fixing a countable dense subset $X_0$ in X, the obvious map $\tilde {X}_x\to \mathbb {R}^X \to \mathbb {R}^{X_0}$ is a continuous injection (as $\tilde {X}_x$ consists of continuous functions), and hence a homeomorphism onto its image. The image is a Frechet space, and thus metrizable. It follows that $\tilde {X}_x$ is metrizable. Since the natural map $\tilde {X}_x\to \bar {X}$ is also a continuous bijection, we conclude that it is a homeomorphism, and deduce that $\bar {X}$ is compact and metrizable.

Proof. The equivariance of the decompositions is obvious. Fix a dense countable subset $X_0$ in X and use the fact that $\tilde {X}$ consists of continuous functions to note that

$$ \begin{align*} \tilde{X}^u=\bigcap_{n\in \mathbb{N}}\bigcup_{x\in X_0} \{ {h\in \tilde{X}}\mid{ h(x)\leq -n } \} , \end{align*} $$

and thus $\tilde {X}^u \subset \tilde {X}$ is measurable. Fixing $x\in X$ , using the measurability of $\tilde {X}^u_x$ , we observe that $\bar {X}^u\subset \bar {X}$ is measurable.

Proof. We will show the analogous statement for $\tilde {X}^b$ rather than $\bar {X}^b$ , and we note that the pseudo-metric we construct below passes to $\bar {X}^b$ (since the sets $\tilde {I}(h)$ below do not change when adding a constant to h).

We first show that the function

$$ \begin{align*} \inf:\tilde{X}^b\to \mathbb{R}, \quad h\mapsto \inf \{ {h(x)}\mid{x\in X} \} \end{align*} $$

is measurable and $\operatorname {Isom}(X)$ -invariant, and that for every $h\in \tilde {X}^b$ , the set

$$ \begin{align*} \tilde{I}(h)= \{ {x\in X}\mid{h(x)< \inf(h)+1} \} \end{align*} $$

is bounded in X.

To see that $\inf $ is measurable, we fix a dense countable subset $X_0$ in X and we use the continuity of the functions in $\tilde {X}^b$ to observe that

$$ \begin{align*} \inf(h)=\inf \{ {h(x)}\mid{x\in X_0} \}. \end{align*} $$

The invariance of this function under $\operatorname {Isom}(X)$ is clear.

Fix $h \in \tilde {X}^b$ . We now argue that $\tilde {I}(h)$ is of diameter bounded by $C=8+4\delta $ , where $\delta $ is the hyperbolicity constant associated with the thin triangle property of X. Without loss of generality, we assume that $\inf (h)=0$ . Assuming the negation, we fix two points $x,x'$ satisfying $d(x,x')>C$ and $h(x),h(x')<1$ . We consider a finite sequence of points $x_0,x_1,\ldots ,x_n$ on a geodesic segment from x to $x'$ such that $x_0=x$ , $x_n=x'$ and $d(x_i,x_{i+1})<1$ . We consider the image of h in $\bar {X}$ along with its neighbourhood given by

$$ \begin{align*} U\kern1.2pt{=}\kern1.2pt \{ {f+\mathbb{R}\cdot\mathbf{1}}\mid{ f\kern1.2pt{\in}\kern1.2pt \mathbb{R}^X~\text{ for all } 0\kern1.2pt{\leq}\kern1.2pt i,j\kern1.2pt{\leq}\kern1.2pt n,~ |(f(x_i)-f(x_j))-(h(x_i)-h(x_j))|\kern1.2pt{<}\kern1.2pt1}\kern-1.2pt \}. \end{align*} $$

We fix a point $y\in X$ whose image in $\bar {X}$ is in U. We thus have

(3.1) $$ \begin{align} \text{for all } 0\leq i,j \leq n, \quad |d(y,x_i)-h(x_i)-d(y,x_j)+h(x_j)|<1. \end{align} $$

We consider geodesic segments from y to x and from y to $x'$ and, using that $x,x'$ and y are the vertices of a thin triangle, we fix i such that $x_i$ lies at distance at most $1+\delta $ from these segments. Thus,

$$ \begin{align*} \begin{aligned} &d(y,x_i)+d(x_i,x)\leq d(y,x)+2+2\delta,\\ &d(y,x_i)+d(x_i,x')\leq d(y,x')+2+2\delta. \end{aligned} \end{align*} $$

Note that $d(x,x_i)+d(x_i,x')=d(x,x')$ . Upon possibly interchanging the roles of x and $x'$ , we will assume that $d(x,x_i)\geq d(x,x')/2$ . In particular, $d(x,x_i)\geq C/2$ . Taking $j=0$ in equation (3.1), we now have

$$ \begin{align*} 0 &= \inf(h) \\ & \leq h(x_i) \\ & < 1+d(y,x_i)+h(x)-d(y,x) \\ &\leq 1+(d(y,x)+2+2\delta- d(x_i,x))+h(x)-d(y,x)\\ &< (3+h(x))+2\delta-d(x_i,x) \\ & \leq 4+2\delta-C/2 =0. \end{align*} $$

This is a contradiction, and thus indeed the diameter of $\tilde {I}(h)$ is bounded by C.

We can now define the required pseudo-metric $\rho $ by setting

$$ \begin{align*} \rho(h,h')=d_{\textrm{Haus}}(\tilde{I}(h), \tilde{I}(h')). \end{align*} $$

This is the pull-back of a metric, so it is a pseudo-metric, and $\Gamma $ clearly acts on it by isometries. The fact that the $\tilde {I}(\cdot )$ are bounded implies the claim on unboundedness of the action. Moreover, coarse separability follows from separability of X and (uniform) boundedness of the $\tilde {I}(\cdot )$ . What is left to show is that $\rho $ is Borel, and to do so, we have to show that for all $r\in \mathbb R$ , we have that $\Delta _r=\{(h,h'): \rho (h,h')> r\}\subseteq \tilde {X}^b \times \tilde {X}^b$ is Borel. Expanding the definitions, $\Delta _r$ is the set of pairs $(h,h')$ such that $\tilde {I}(h)$ and $\tilde {I}(h')$ lie at Hausdorff distance more than r. In turn, this means that there exists $x\in \tilde {I}(h)\cap X_0$ such that all $y\in \tilde {I}(h')\cap X_0$ lie at distance more than r from x, or that the same holds switching the roles of h and $h'$ (to justify the ‘ $\cap X_0$ ’, note that $\tilde {I}(h)\cap X_0$ is dense in $\tilde {I}(h)$ in view of the continuity of h, and similarly for $h'$ ). Denoting $\tilde {J}(z)=\{k\in \tilde {X}^b: z\in \tilde {I}(k)\}$ for $z\in Z$ , the first case is further equivalent to $h'$ not lying in $\bigcup _{y\in X_0 : d(x,y)\leq r} \tilde {J}(y)$ , which is the subset of $\tilde {X}^b$ of all k such that $\tilde {I}(k)$ contains some $y\in X_0$ within distance r of x (the other case is similar). Hence, we have

$$ \begin{align*} \Delta_r&=\bigcup_{x\in X_0}\bigg( \tilde{J}(x) \times \bigg(\tilde{X}^b-\bigcup_{y\in X_0 : d(x,y)\leq r}\tilde{J}(y)\bigg) \bigg)\\ &\quad\cup \bigcup_{y\in X_0} \bigg( \bigg(\tilde{X}^b-\bigcup_{x\in X_0 : d(x,y)\leq r} \tilde{J}(y)\bigg)\times \tilde{J}(x)\bigg). \end{align*} $$

Hence, $\Delta _r$ is Borel provided that all $\tilde J(z)$ are Borel. However, $\tilde J(z)=\{k\in \tilde {X}^b: k(z)<\inf (k)+1\}$ , so it is measurable in view of the measurability of $\inf $ , as required.

Proof. We will denote the lift of $\bar {h}$ in $\tilde {X}_o$ by h and show that $(x_n)$ converges to infinity if and only if $h\in \tilde {X}^u_o$ . Note that for every $x\in X$ , we have $d(x_n,x)-|x|\to h(x)$ .

Assuming first $(x_n)$ converges to infinity, we will show that $h\in \tilde {X}^u_o$ . Fix $r>0$ . Fix N such that for $n,m>N$ , $(x_n,x_m)>r$ . Fix $m>N$ , note that $|x_m|\geq r$ and let x be a point on a geodesic segment from o to $x_m$ with $|x|=r$ . Then, by hyperbolicity,

$$ \begin{align*} (x_n,x) \geq \min\{(x_n,x_m),(x_m,x)\}-\delta = r-\delta. \end{align*} $$

Thus,

$$ \begin{align*} h(x)=\lim_{n\to \infty} (d(x_n,x)-|x_n|)= \lim_{n\to \infty} (|x|-2(x_n,x)) \leq 2\delta-r. \end{align*} $$

As r was arbitrary, we get indeed that $h\in \tilde {X}^u_o$ .

Assuming now $h\in \tilde {X}^u$ , we will show that $(x_n)$ converges to infinity. Fix $r>0$ . Fix x such that $h(x)<-r$ . Fix N such that for every $n>N$ , $d(x_n,x)-|x_n|<-r$ and observe that for such n,

$$ \begin{align*} (x_n,x)=\tfrac{1}{2}(|x_n|+|x|-d(x_n,x))\geq -\tfrac{1}{2}(d(x_n,x)-|x_n|)> \tfrac{1}{2} r. \end{align*} $$

Then, by hyperbolicity, for $n,m>N$ ,

$$ \begin{align*} (x_n,x_m) \geq \min\{(x_n,x),(x,x_m)\}-\delta> \tfrac{1}{2} r-\delta. \end{align*} $$

As r was arbitrary, we deduce indeed that the sequence $(x_n)$ converges to infinity.

In the setting of the former paragraph, if $(x^{\prime }_n)$ is another sequences in X satisfying $x^{\prime }_n \to \bar {h}$ , fixing $N'\geq N$ such that for every $n>N'$ , $d(x^{\prime }_n,x)-|x^{\prime }_n|<-r$ , the same computation shows that $(x_n,x^{\prime }_n)> r/2-\delta $ . Thus indeed, we obtain $(x_n,x^{\prime }_n) \to \infty $ .

Proof. The equivariance of $\pi $ is obvious. To show continuity, we fix a point $\bar {h}\in \bar {X}^u$ and show the continuity of $\pi $ at $\bar {h}$ . Thus, we fix $r>0$ and argue to show that there exists a neighbourhood V of $\bar {h}$ in $\bar {X}^u$ such that $\pi (V)\subset U(\pi (\bar {h}),r)$ . We will denote the lift of $\bar {h}$ in $\tilde {X}^u_o$ by h, set $t=2(r+\delta )$ and fix a point $x\in X$ such that $h(x)<-t$ . We let $V\subset \bar {X}^u_o$ be the open neighbourhood of $\bar {h}$ corresponding to the set $\{h'\in \tilde {X}^u_o \mid h'(x)<-t\}$ . Fix $\bar {h}'\in V$ and denote its lift in $\tilde {X}_o^u$ by $h'$ . Let $(x_n)$ and $(x^{\prime }_n)$ be sequences in X converging to $\bar {h}$ and $\bar {h}'$ , respectively. In particular, we have

$$ \begin{align*} h(x) =\lim_{n\to\infty} (d(x_n,x)-|x_n|) \quad \text{and} \quad h'(x) =\lim_{n\to\infty} (d(x^{\prime}_n,x)-|x^{\prime}_n|). \end{align*} $$

Fix N such that for every $n>N$ , both $(d(x_n,x)-|x_n|)<-t$ and $(d(x^{\prime }_n,x)-|x^{\prime }_n|)<-t$ . Note that for $n>N$ ,

$$ \begin{align*} (x_n,x)=\tfrac{1}{2}(|x_n|+|x|-d(x_n,x))\geq -\tfrac{1}{2} (d(x_n,x)-|x_n|)> \tfrac{1}{2} t \end{align*} $$

and similarly $(x^{\prime }_n,x)>t/2$ . Thus, we have

$$ \begin{align*} (x_n,x^{\prime}_n)\geq \min\{(x_n,x),(x,x^{\prime}_n)\}-\delta> \tfrac{1}{2} t-\delta=r. \end{align*} $$

It follows that $\liminf (x_n,x^{\prime }_n)\geq r$ and, in particular,

$$ \begin{align*} \sup \big\{ {\liminf_{n\to\infty} (x_n,x^{\prime}_n)}\mid{(x_n)\in\pi(\bar{h}),~(x^{\prime}_n)\in \pi(\bar{h}')} \big\} \geq r. \end{align*} $$

Therefore, $\pi (\bar {h}')\in U(\pi (\bar {h}),r)$ and we conclude that indeed $\pi (V)\subset U(\pi (\bar {h}),r)$ .

Proof. Denote $\mathcal T=\partial X^{(3)}$ for convenience. Fix a $\Gamma $ -invariant dense subset A of X.

We claim that there exists a $\Gamma $ -equivariant map $\tau :\mathcal T\to \operatorname {Bdd}(A)$ with the property that for any $x\in X$ , we have that $\mathcal T_x=\tau ^{-1}(\{B\in \operatorname {Bdd}(X): x\in B\})\subseteq \mathcal T$ is closed.

To show the claim, we will in fact first define a closed subset $\mathcal T_x$ for each $x\in A$ and then set $\tau (t)=\{x\in A: t\in \mathcal T_x\}$ . Let $\delta>0$ be a hyperbolicity constant for X. For $x\in A$ , let $T_x\subseteq X^3$ be the set of all triples $(x_1,x_2,x_3)$ such that ${d(x_i,x_j)> d(x_i,x)+d(x,x_j)-10\delta }$ for all distinct $i,j\in \{1,2,3\}$ . We then define $\mathcal T_x$ to be the intersection of $\mathcal T$ with closure of $T_x$ (taken in $(X\cup \partial X)^3$ ). Notice that $\tau (t)$ defined as above is then a bounded, non-empty subset of A, and that $\tau $ is $\Gamma $ -equivariant.

We can now pull-back the Hausdorff distance $d_{\textrm {Haus}}$ on $\operatorname {Bdd}(A)$ to $\mathcal T$ , that is, we set $\rho (t,t')=d_{\textrm {Haus}}(\tau (t),\tau (t'))$ . Clearly, $\rho $ is a pseudo-metric and $\Gamma $ acts on $\mathcal T$ by isometries of $\rho $ . Moreover, orbits are unbounded since $(\mathcal T,\rho )$ is $\Gamma $ -equivariantly isometrically embedded in $\operatorname {Bdd}(A)$ , and on the latter, the $\Gamma $ -orbits are unbounded since orbits are unbounded in X (a similar argument applies to subsets of $\Gamma $ ). Coarse separability also arises from this isometric embedding together with separability of X, which gives coarse separability of $\operatorname {Bdd}(A)$ . It remains to show that $\rho $ is Borel. To do so, it suffices to show that for all $r\in \mathbb R$ , we have that $\Delta _r=\{(t,t'): \rho (t,t')> r\}\subseteq \mathcal T \times \mathcal T$ is Borel. Expanding the definitions, $\Delta _r$ is the set of pairs $(t,t')$ such that $\tau (t)$ and $\tau (t')$ lie at Hausdorff distance more than r. In turn, this means that there exists $x\in \tau (t)$ such that all $y\in \tau (t')$ lie at distance more than r from x, or that the same holds switching the roles of t and $t'$ . The first case is further equivalent to $t'$ not lying in $\bigcup _{y\in A : d(x,y)\leq r} \mathcal T_y$ , so that $\tau (t)$ does not contain any y with $d(x,y)\leq r$ (and the other case is similar). Hence, we have

$$ \begin{align*}\Delta_r=\bigcup_{x\in A}\bigg( \mathcal T_x \times \bigg(\mathcal T-\bigcup_{y\in A : d(x,y)\leq r} \mathcal T_y\bigg) \bigg)\cup\bigcup_{y\in A} \bigg( \bigg(\mathcal T-\bigcup_{x\in A : d(x,y)\leq r} \mathcal T_y\bigg)\times \mathcal T_x\bigg).\end{align*} $$

Since all $\mathcal T_x$ are closed, $\Delta _r$ is Borel, as required.

Proof. We have a $\Gamma $ -map

$$ \begin{align*} \Phi:\operatorname{Prob}_c(\partial X) \longrightarrow \operatorname{Prob}(\partial X^3),\quad \mu\mapsto \mu\times\mu\times\mu. \end{align*} $$

In fact, the assumption that $\mu $ has no atoms on a space $\partial X$ implies that $\mu \times \mu \times \mu $ gives zero mass to the diagonals in $\partial X \times \partial X\times \partial X$ , and so is fully supported on $\partial X^{(3)}$ . We thus get a well-defined map

$$ \begin{align*} \operatorname{Prob}_c(\partial X) \longrightarrow \operatorname{Prob}(\partial X^{(3)}),\quad \mu\mapsto \mu\times\mu\times\mu, \end{align*} $$

as required.

To prove the moreover part, let $\mu ,\nu \in \operatorname {Prob}_c(\partial X)$ and let $f\in L^\infty (\partial X)$ be the Radon–Nikodym derivative of $\mu $ with respect to $\nu $ . Then $(x,y,z)\mapsto f(x)f(y)f(z)$ is the Radon–Nikodym derivative of $\Phi (\mu )$ with respect to $\Phi (\nu )$ , so that $\|{d\Phi (\mu )}/{d\Phi (\nu )}\|_\infty = (\|{d\mu }/{d\nu }\|_\infty )^3$ . The measures $\Psi (\mu )$ and $\Psi (\nu )$ are push-forward measures of $\Phi (\mu )$ and $\Phi (\nu )$ , and push-forwards do not increase the $L^\infty $ norm of Radon–Nikodym derivatives, so we obtain the required bound.

Claim 3.9. We have

$$ \begin{align*} \operatorname{Map}_\Gamma(B_-,\operatorname{Prob}(\bar X^b)) & =\operatorname{Map}_\Gamma(B_+,\operatorname{Prob}(\bar X^b))\\ &=\operatorname{Map}_\Gamma(B_-\times B_+,\operatorname{Prob}(\partial X^{(3)}))\\ &=\varnothing. \end{align*} $$

Proof. Recall Lemmas 3.4 and 3.7 about the existence of Borel coarsely separable pseudo-metrics on $\bar X^b$ and on $\partial X^{(3)}$ , respectively, on which $\Gamma $ acts by isometries with unbounded orbits. Then, in view of Proposition 2.15, all sets of $\Gamma $ -maps in the statement need to be empty (note that any pseudo-metric is a coarse metric).

Claim 3.10. Let $\Gamma \curvearrowright \Omega $ be an ergodic action and $\Gamma \curvearrowright V$ be a Borel action on a Polish space. Suppose $V=V_0\cup V_1$ with $V_0\cap V_1=\varnothing $ is a decomposition into $\Gamma $ -invariant Borel sets and suppose $\operatorname {Map}_\Gamma (\Omega ,\operatorname {Prob}(V_0))=\varnothing $ . Then the inclusion $\operatorname {Prob}(V_1)\subset \operatorname {Prob}(V)$ gives an identification

$$ \begin{align*} \operatorname{Map}_\Gamma(\Omega,\operatorname{Prob}(V))=\operatorname{Map}_\Gamma(\Omega,\operatorname{Prob}(V_1)). \end{align*} $$

Proof. Consider a measurable $\Gamma $ -equivariant map $\phi \colon \Omega \longrightarrow \operatorname {Prob}(V)$ . We also define $f_0,f_1\colon \Omega \to [0,1]$ by setting $f_i(\omega )=\phi (\omega )(V_i)$ . These are $\Gamma $ -invariant measurable functions. By ergodicity, $f_i(\omega )=c_i$ constants and $c_i\ge 0$ with $c_0+c_1=1$ . Assuming $c_0>0$ , one obtains a contradiction, because the normalized restriction $\phi _0(\omega )=c_0^{-1}\cdot \phi (\omega )|_{V_0}$ is a $\Gamma $ -map to $\operatorname {Prob}(V_0)$ , which are ruled out by assumption. Hence, $c_0=0$ and $\phi (\omega )$ gives full mass to $V_1$ , and hence $\phi $ can be considered as a $\Gamma $ -map $\Omega \longrightarrow \operatorname {Prob}(V_1)$ .

Claim 3.11. For all $\psi _-\in \operatorname {Map}(B_-,\operatorname {Prob}(\partial X))$ and $\psi _+\in \operatorname {Map}(B_+,\operatorname {Prob}(\partial X))$ , the diagonal $\Delta = \{ {(t,t)}\mid {t\in \partial X} \} \subset \partial X^2$ satisfies

$$ \begin{align*} \psi_-(x)\times \psi_+(y)(\Delta)=0 \end{align*} $$

for a.e. $(x,y)\in B_-\times B_+$ .

Proof. Given a probability measure $\eta \in \operatorname {Prob}(\partial X)$ , denote by $\operatorname {Atom}(\eta )$ the set of atoms of $\eta $ ; and for $\epsilon>0$ , by $\operatorname {Atom}_\epsilon (\eta )= \{ {a}\mid {\eta (\{a\})\ge \epsilon } \} $ the subset of atoms with weight $\ge \epsilon $ . The cardinality of $\operatorname {Atom}_\epsilon (\eta )$ is bounded by $\lceil 1/\epsilon \rceil $ and $\operatorname {Atom}(\eta )$ is at most countable. Let $\alpha ,\beta \in \operatorname {Prob}(\partial X)$ be two probability measures. A standard application of Fubini’s theorem shows that

$$ \begin{align*} \alpha\times\beta(\Delta)=\sum_{t\in \partial X} \alpha(\{t\})\cdot\beta(\{t\}), \end{align*} $$

where the non-zero terms correspond to $t\in \operatorname {Atom}(\alpha )\cap \operatorname {Atom}(\beta )$ .

For $(x,y)\in B_-\times B_+$ , observe that the sets $\operatorname {Atom}(\psi _-(x))$ , $\operatorname {Atom}(\psi _+(y))$ and their intersection vary measurably and $\Gamma $ -equivariantly.

Assume, towards contradiction, that $\psi _-(x)\times \psi _+(y)(\Delta )>0$ . This is equivalent to asserting that for some $\epsilon>0$ , the set

$$ \begin{align*} \{ {(x,y)\in B_-\times B_+}\mid{\operatorname{Atom}_\epsilon(\psi_-(x))\cap\operatorname{Atom}_\epsilon(\psi_+(y))\ne \varnothing} \} \end{align*} $$

has positive measure. Since the $\Gamma $ -action on $B_-\times B_+$ is ergodic, we deduce that this set is conull, and the cardinality of the set $A(x,y)=\operatorname {Atom}_\epsilon (\psi _-(x))\cap \operatorname {Atom}_\epsilon (\psi _+(y))$ is essentially constant.

By Fubini’s theorem for a.e. $x\in B_-$ , we have $A(x,y)=A(x,y')$ for a.e. $y,y'\in B_+$ . Thus, $A(x,-)$ is essentially independent of the second variable. For the same reason, it is essentially independent of the first variable; hence, $A(x,y)=A$ for a fixed finite subset $A\subset \partial X$ . Since the map $(x,y)\mapsto A(x,y)=A$ is equivariant, it follows that $\Gamma $ has a finite orbit O on $\partial X$ . By hypothesis, $\Gamma $ does not fix a point or a pair of points in $\partial X$ , so that $|O|\geq 3$ . By considering the restriction of the pseudo-metric on $\partial X^{(3)}$ from Lemma 3.7 to the finite set of triples of distinct points in O, we obtain a $\Gamma $ -invariant orbit in $\partial X^{(3)}$ , contradicting that the action on $\partial X^{(3)}$ has unbounded orbits.

Hence, $\psi _-(x)\times \psi _+(y)(\Delta )=0$ almost everywhere on $B_-\times B_+$ , as claimed.

End of proof of Theorem 3.1.

Since the actions $\Gamma \curvearrowright B_-$ , $\Gamma \curvearrowright B_+$ are amenable, the spaces $\operatorname {Map}_\Gamma (B_\pm ,\operatorname {Prob}(\bar X))$ are not empty. Consider the $\Gamma $ -invariant Borel decomposition $\bar {X}=\bar {X}^b\cup \bar {X}^u$ . In view of Claims 3.9 and 3.10, we have

$$ \begin{align*} \operatorname{Map}_\Gamma(B_\pm,\operatorname{Prob}(\bar{X}))=\operatorname{Map}_\Gamma(B_\pm,\operatorname{Prob}(\bar{X}^u)) \end{align*} $$

and post-composing with the $\Gamma $ -map $\pi _*:\operatorname {Prob}(\bar X^u)\to \operatorname {Prob}(\partial X)$ (see §3.3), we obtain

$$ \begin{align*} \operatorname{Map}_\Gamma(B_\pm,\operatorname{Prob}(\partial X))\ne \varnothing. \end{align*} $$

Let us choose some measurable $\Gamma $ -equivariant maps

$$ \begin{align*} \psi_{-}:B_{-}\longrightarrow \operatorname{Prob}(\partial X),\quad \psi_{+}:B_{+}\longrightarrow \operatorname{Prob}(\partial X). \end{align*} $$

Consider the partition of $(\partial X)^3$ into $\Gamma $ -invariant Borel sets

$$ \begin{align*} (\partial X)^3=\partial X^{(3)}\cup \Delta_{12}\cup \Delta^{\prime}_{13}\cup \Delta^{\prime}_{23}, \end{align*} $$

where $\Delta _{ij}= \{ {(\xi _1,\xi _2,\xi _3)}\mid {\xi _i=\xi _j} \} $ , $\Delta _{123}= \{ {(\xi ,\xi ,\xi )}\mid {\xi \in \partial X} \} $ , $\Delta ^{\prime }_{ij}=\Delta _{ij}\setminus \Delta _{123}$ .

Note that $\operatorname {Map}_\Gamma (B_-\times B_+,\operatorname {Prob}(\partial X^{(3)}))=\varnothing $ by Claim 3.9. Therefore, using Claim 3.10, every measurable $\Gamma $ -map $\Psi :B_{-}\times B_{+}\longrightarrow \operatorname {Prob}((\partial X)^3)$ gives full measure to $\Delta _{12}\cup \Delta ^{\prime }_{13}\cup \Delta ^{\prime }_{23}$ . We shall apply this to the map

$$ \begin{align*} \Psi(x,y)=\psi_-(x)\times \psi_+(y)\times \tfrac{1}{2}(\psi_-(x)+\psi_+(y)). \end{align*} $$

Note that Claim 3.11 implies that almost everywhere, $\Psi (x,y)(\Delta _{12})=0$ .

Let $\Delta $ denote the diagonal in $\partial X \times \partial X$ . The projection $\Delta ^{\prime }_{13}\to \Delta $ , $(\xi ,\eta ,\xi )\mapsto (\xi ,\xi )$ shows that

$$ \begin{align*} \Psi(x,y)(\Delta^{\prime}_{13})\le\Psi(x,y)(\Delta_{13})=\tfrac{1}{2}\psi_-(x)\times\psi_+(y)(\Delta) +\tfrac{1}{2}\psi_-(x)\times\psi_-(x)(\Delta). \end{align*} $$

Moreover, the first term of the right-hand side vanishes by Claim 3.11, so that $\Psi (x,y)(\Delta ^{\prime }_{13})\le \tfrac 12\psi _-(x)\times \psi _-(x)(\Delta )$ . Similarly, using the projection $\Delta ^{\prime }_{23}\to \Delta $ , $(\eta ,\xi ,\xi )\mapsto (\xi ,\xi )$ gives

$$ \begin{align*} \Psi(x,y)(\Delta^{\prime}_{23})\le\Psi(x,y)(\Delta_{23})=\tfrac{1}{2}\psi_+(y)\times\psi_+(y)(\Delta). \end{align*} $$

We conclude that

$$ \begin{align*} 1 &= \Psi(x,y)(\Delta^{\prime}_{13})+ \Psi(x,y)(\Delta^{\prime}_{23}) \\ &\leq \tfrac{1}{2}\psi_-(x)\times\psi_-(x)(\Delta) + \tfrac{1}{2}\psi_+(y)\times\psi_+(y)(\Delta) \\ &\leq 1, \end{align*} $$

so that $\psi _-(x)\times \psi _-(y)(\Delta )=\psi _+(y)\times \psi _+(y)(\Delta )=1$ . Recall that, by Fubini’s theorem, for any probability measure $\nu $ , one has $\nu \times \nu (\Delta )=1$ if and only if $\nu $ is a Dirac measure $\nu =\delta _z$ for some z. Hence, there exist maps $\phi _\pm :B_\pm \longrightarrow \partial X$ so that

$$ \begin{align*} \psi_-(x)=\delta_{\phi_-(x)},\quad \psi_+(y)=\delta_{\phi_+(y)}. \end{align*} $$

The maps $\phi _\pm :B_{\pm} \longrightarrow \partial X$ are measurable and $\Gamma $ -equivariant because the maps $\psi _\pm :B_\pm \longrightarrow \operatorname {Prob}(\partial X)$ are.

We also note that, since $\Psi (\Delta _{12})=0$ almost everywhere, we have $\phi _-(x)\ne \phi _+(y)$ for a.e. $(x,y)\in B_-\times B_+$ . Therefore, the map $\phi _{\bowtie }=\phi _{-}\times \phi _+$ can be viewed as a measurable $\Gamma $ -equivariant map $B_-\times B_+ \longrightarrow \partial X^{(2)}$ into distinct pairs.

Next, we show that $\phi _\pm $ are essentially unique. Indeed, we showed that every $\psi _\pm \in \operatorname {Map}_\Gamma (B_\pm ,\operatorname {Prob}(\partial X))$ takes values in Dirac measures. Assume we have two pairs of $\Gamma $ -maps $\psi ^i_\pm :B_\pm \to \operatorname {Prob}(\partial X)$ with $i=1,2$ . Consider the $\Gamma $ -maps $B_{\pm }\longrightarrow \operatorname {Prob}(\partial X)$ given by $z\mapsto \tfrac 12(\psi ^1_\pm (z)+\psi ^2_\pm (z))$ . Such a map has the form $z\mapsto \delta _{\phi _\pm (z)}$ only if $\psi ^1_\pm (z)=\psi ^2_\pm (z)=\delta _{\phi _\pm (z)}$ .

Next, consider an arbitrary map $\theta \in \operatorname {Map}_\Gamma (B_-\times B_+,\partial X)$ , consider the measurable $\Gamma $ -equivariant map to be defined by

$$ \begin{align*} \Theta:B_-\times B_+ \longrightarrow (\partial X)^3, \quad\Theta(x,y)=(\theta(x,y),\phi_-(x),\phi_+(y)). \end{align*} $$

Using the $\Gamma $ -equivariant Borel decomposition $(\partial X)^3=\partial X^{(3)}\cup \Delta ^{\prime }_{12}\cup \Delta ^{\prime }_{13}\cup \Delta _{23}$ and ergodicity of the $\Gamma $ -action on $B_-\times B_+$ , we deduce that $\Theta (x,y)$ lies in one of these sets for a.e. $(x,y)$ . Lemma 3.7 combined with coarse metric ergodicity of $B_-\times B_+$ rules out the set of distinct triples $\partial X^{(3)}$ . Since the $23$ -components of $\Theta $ is just $\phi _{\bowtie }$ , and the latter take values in the space of distinct pairs $\partial X^{(2)}$ , rule out $\Delta _{23}$ . It follows that, up to a null set, either $\Theta (x,y)\in \Delta ^{\prime }_{12}$ or $\Theta (x,y)\in \Delta ^{\prime }_{13}$ . This corresponds to $\theta =\psi _-\circ \operatorname {pr}_-$ or $\theta =\psi _+\circ \operatorname {pr}_+$ as claimed in assertion (ii).

Finally, notice that assertion (iii) can be deduced from assertion (ii) by looking at the coordinates in $(\partial X)^{(2)}$ .

Claim 4.5. The map $\phi $ factors through one of $B_i$ . In other words, there is an index $i\in \{1,\ldots ,N\}$ and a $\Gamma $ -map $B_i\ {\buildrel {\phi _i}\over \longrightarrow}\ \partial X$ such that

$$ \begin{align*} \phi\colon B\ {\buildrel{\operatorname{pr}_i}\over\longrightarrow}\ B_i\ {\buildrel{\phi_i}\over\longrightarrow}\ \partial X. \end{align*} $$

Proof. The equivariant map $\phi :B \longrightarrow \partial X$ cannot be essentially constant, because $\Gamma $ does not have a fixed point in $\partial X$ . Fix an $i\in \{1,\ldots ,N\}$ so that $\phi $ is not essentially constant over the $B_i$ -factor of $B=B_1\times \cdots \times B_N$ . We let $B'$ be the product of the other factors, and identify $B\simeq B_i\times B'$ . Using this identification, we consider the map

$$ \begin{align*} \Phi:B_i\times B' \times B_i\times B' \to \partial X^2,\quad (x,y,x',y') \mapsto (\phi(x,y),\phi(x,y')). \end{align*} $$

By Theorem 3.1(iii), we have three cases: $\Phi (B\times B)$ is contained in the diagonal $\Delta \subset \partial X^2$ , $\Phi =\phi _{\bowtie }$ or $\Phi =\tau \circ \phi _{\bowtie }$ , where $\phi _{\bowtie }=\phi \times \phi $ and $\tau (\xi ,\eta )=(\eta ,\xi )$ . In the first case, we see that $\phi (x,y)$ is independent of $y\in B'$ and therefore descends to a $\Gamma $ -map $B_i\to \partial X$ as required. In the second and third cases, $\phi $ is independent of $x\in B_i$ , contradicting our choice of i.

Claim 4.6. There exist $L,C$ so that for all $\gamma ,\gamma '$ , we have

$$ \begin{align*} d(\gamma,\gamma')\le L\cdot d_i(\gamma,\gamma')+C. \end{align*} $$

Proof. As G is of higher rank, we deduce from §4.4 that $N>1$ and, in particular, we get that $\operatorname {pr}_i(\Gamma )$ is dense in $G_i$ . Hence, in the metric $d_i$ any pair of points is connected by a $(1,1)$ -quasi-geodesic, and to prove the claim, it suffices to show that sequences that are bounded in $G_i$ are bounded in $(\Gamma ,d)$ . In other words, it suffices to show that any sequence $\{\gamma _j\}$ in $\Gamma $ for which $\{\operatorname {pr}_i(\gamma _j)\}$ is precompact in $G_i$ , one has

$$ \begin{align*} \sup_j d(\gamma_j,1)<+\infty. \end{align*} $$

Let $\mu =\phi _*\nu _i\in \operatorname {Prob}(\partial X)$ be the push-forward of the measure $\nu _i$ on $B_i$ . By the metric ergodicity of $B_i$ , $\mu $ has no atomic part. Indeed, if it had, we would get a countable invariant subset of $\partial X$ and upon endowing it with the discrete metric, in view of the metric ergodicity of $B_i$ , we will conclude that this set contains a single point which is $\Gamma $ -invariant, which contradicts the hypothesis.

By Lemma 3.8, we have a $\Gamma $ -map

$$ \begin{align*} \Psi:\operatorname{Prob}_c(\partial X) \longrightarrow \operatorname{Prob}(\partial X^{(3)}), \end{align*} $$

where, as before, $\operatorname {Prob}_c(\partial X)$ is the set of all atom-less probability measures on $\partial X$ .

We assume that $\{\gamma _j\}$ is such that $\{\operatorname {pr}_i(\gamma _j)\}$ is precompact in $G_i$ . We now need the following lemma.

Lemma 4.7. Given any precompact subset $\{\gamma _j\}\subseteq G_i$ , the corresponding Radon–Nikodym derivatives are uniformly bounded, that is, we have

$$ \begin{align*} \sup_{j\ge 1} \bigg\|\frac{d\gamma_j\nu_i}{d\nu_i}\bigg\|_\infty<+\infty. \end{align*} $$

Proof. Since $\nu _i$ is the Patterson–Sullivan measure for $G_i$ , we have for any $g\in G_i$ and a.e. $\xi \in \partial X_i$ ,

$$ \begin{align*} \frac{d g\nu_i}{d\nu_i}(\xi)=e^{-\delta_i \beta(g,\xi)}, \end{align*} $$

where $\delta _i$ is a constant associated with the symmetric space $X_i$ for $G_i$ and $\beta (g,\xi )$ is the Busemann function. Since the Busemann function is bounded by the distance in $X_i$ , $|\beta (g,\xi )|\le d_i(go,o)$ , the lemma is proved.

Hence, by Lemma 4.7, we have

$$ \begin{align*} \sup_{j\ge 1} \bigg\|\frac{d\gamma_j\nu_i}{d\nu_i}\bigg\|_\infty=C<+\infty. \end{align*} $$

We thus have the same bound on the Radon–Nikodym derivatives of $\mu $

$$ \begin{align*} \sup_{j\ge 1} \bigg\|\frac{d\gamma_j\mu}{d\mu}\bigg\|_\infty\le C \end{align*} $$

and also, by the ‘moreover’ part of Lemma 3.8,

$$ \begin{align*} \sup_{j\ge 1} \bigg\|\frac{d\Psi(\gamma_j\mu)}{d\Psi(\mu)}\bigg\|_\infty\le C^3. \end{align*} $$

Setting $U=\operatorname {Prob}(\partial X^{(3)})$ , recall the maps $\beta _\epsilon :\operatorname {Prob}(\partial X^{(3)})\to \operatorname {Bdd}(\partial X^{(3)})$ from §2.6, which are $\Gamma $ -equivariant (as they are equivariant with respect to all isometries). Fix ${\epsilon \in (0,1/2)}$ and set $R=r_{\epsilon /C}(\Psi (\mu ))+1$ . Fixing j and applying Lemma 2.16 with $m_1=\Psi (\mu )$ and $m_2=\Psi (\gamma _j\mu )$ , we get

$$ \begin{align*} \gamma_j (\beta_{\epsilon}\circ \Psi(\mu))=\beta_{\epsilon}\circ\Psi(\gamma_j\mu) \subset N_R(\beta_{\epsilon/C}(\Psi(\mu))). \end{align*} $$

Since $N_R(\beta _{\epsilon /C}(\Psi (\mu )))$ is a bounded subset of $\partial X^{(3)}$ , we get that the sequence $\{\gamma _j\}$ has bounded orbits in $\partial X^{(3)}$ , whence it is bounded in $(\Gamma ,d)$ by the ‘moreover’ part of Lemma 3.7.

Claim 4.8. There exist $L,C$ so that for all $\gamma ,\gamma '$ , we have

$$ \begin{align*} d_i(\gamma,\gamma')\le L\cdot d(\gamma,\gamma')+C. \end{align*} $$

Proof. Let $\gamma _0\in \Gamma $ be a loxodromic element for $\Gamma \curvearrowright X$ , which exists by Gromov’s classification of actions on hyperbolic spaces [Reference Gromov11, §8]. Since the identity map $(\Gamma ,d_i)\to (\Gamma ,d)$ is coarsely Lipschitz, $\gamma _0$ is loxodromic for $\Gamma \curvearrowright X_i$ as well. Let us choose a constants A so that for each $j\geq 0$ , we have $j/A\leq d_i(1,\gamma _0^j)\leq Aj$ and $j/A\leq d(1,\gamma _0^j)\leq Aj$ . Note that, up to increasing A, since $G_i$ is rank-one and $\gamma _0$ is loxodromic, the set

$$ \begin{align*} \{\kappa\gamma_0^j\mid \kappa\in K_i,~j\geq 0\} \end{align*} $$

is A-dense in $X_i$ , where we denote by $K_i$ a maximal compact subgroup of $G_i$ . This is because, since $K_i$ acts transitively on $\partial X$ , there exists a constant C such that given any two points on a sphere around $x_i$ , there is an element of $K_i$ that moves the first point C-close to the second one.

Since the rank of $G_i$ equals one, we know by hypothesis that $N>1$ and, in particular, we get that $\operatorname {pr}_i(\Gamma )$ is dense in $G_i$ . Approximating elements of $K_i$ by elements of $\Gamma $ , we get that the set

$$ \begin{align*} \{\kappa\gamma_0^j\mid d_i(1,\kappa)\leq 1, j\geq 0\} \end{align*} $$

is $A+1$ -dense in $\Gamma $ for the metric $d_i$ . Enlarging A if necessary, let us further assume that $d(1,\gamma )\leq Ad_i(1,\gamma )+A$ for all $\gamma \in \Gamma $ .

Consider an arbitrary $\gamma \in \Gamma $ . Using the above, we find $\kappa \in \Gamma $ such that $d_i(1,\kappa )\leq 1$ and $d_i(\kappa \gamma ,\gamma _0^j)\leq A+1$ for some $j\geq 0$ . Therefore, we obtain

$$ \begin{align*} d_i(1,\gamma) &= d_i(\kappa,\kappa\gamma)\\ &\leq d_i(1,\gamma_0^j)+A+2\\ & \leq Aj+A+2\\ & \leq A^2d(1,\gamma_0^j)+A+2\\ &\leq A^2 (d(1,\kappa)+d(\kappa,\kappa\gamma)+d(\kappa\gamma,\gamma_0^j))+A+2\\ & \leq A^2d(1,\gamma)+A^3+2A^2+3A+2, \end{align*} $$

as required.