1 Introduction
The critical exponent of a discrete group of isometries
$\Gamma $
of a proper metric space X is defined as
$$ \begin{align*} h_\Gamma := \limsup_{T\to +\infty}\frac{1}{T}\log \Gamma x \cap \overline{B}(x,T), \end{align*} $$
where x is any point of X. In [Reference CavallucciCav21b] the author proved that if X is a Gromov-hyperbolic space then the limit superior above is a true limit (see also Lemma 4.12).
A discrete group
$\Gamma $
of isometries of a proper,
$\delta $
-hyperbolic metric space is said to be quasiconvex-cocompact if it acts cocompactly on the quasiconvex hull of its limit set
$\Lambda (\Gamma )$
, namely QC-Hull
$(\Lambda (\Gamma ))$
. In this case the codiameter is by definition the diameter of the quotient metric space
$\Gamma \backslash $
QC-Hull
$(\Lambda (\Gamma ))$
.
In the sequel we denote by
$\mathcal {M}(\delta , D)$
the class of triples
$(X,x,\Gamma )$
where X is a proper,
$\delta $
-hyperbolic metric space,
$\Gamma $
is a discrete, torsion-free, non-elementary, quasiconvex-cocompact group of isometries of X with codiameter less than or equal to D and x is a point of QC-Hull
$(\Lambda (\Gamma ))$
. We refer to §4 for the details of all these definitions.
We are interested in convergence of sequences of triples in
$\mathcal {M}(\delta ,D)$
in the equivariant pointed Gromov–Hausdorff sense, as defined by Fukaya in [Reference FukayaFuk86]. This is a version of the classical pointed Gromov–Hausdorff convergence that considers also the groups acting on the spaces. Its precise definition is recalled in §3. Our main result is the following theorem.
Theorem A. Let
$\delta , D\geq 0$
and let
$(X_n,x_n,\Gamma _n)_{n\in \mathbb {N}} \subseteq \mathcal {M}(\delta ,D)$
. If the sequence
$(X_n,x_n,\Gamma _n)$
converges in the equivariant pointed Gromov–Hausdorff sense to
$(X_\infty , x_\infty , \Gamma _\infty )$
then:
-
(i)
$(X_\infty , x_\infty , \Gamma _\infty ) \in \mathcal {M}(\delta ,D)$
; and -
(ii)
$h_{\Gamma _\infty } = \lim _{n\to +\infty }h_{\Gamma _n}$
.
The first difficulty in the proof of (i) is to show that the limit group
$\Gamma _\infty $
is discrete. The proof is based on a result of [Reference Besson, Courtois, Gallot and SambusettiBCGS21]: if
$(X,x,\Gamma )\in \mathcal {M}(\delta ,D)$
satisfies
$h_\Gamma \leq H < +\infty $
, then the global systole of
$\Gamma $
is bigger than some positive constant depending only on
$\delta ,D$
and H (cf. Proposition 5.3). This is a powerful tool when used together with Corollary 5.9: under the assumptions of Theorem A the critical exponents of the groups
$\Gamma _n$
are uniformly bounded above by some
$H<+\infty $
. All the assumptions on the class
$\mathcal {M}(\delta ,D)$
are necessary in order to get the discreteness of the limit group; see §8. The second difficulty is to show that
$\Gamma _\infty $
is quasiconvex-cocompact. In order to do so we will show that the limit of the Gromov boundaries
$\partial X_n$
can be seen as a canonical subset of
$\partial X_\infty $
; see Proposition 5.11. Under this identification the limit of the sets
$\Lambda (\Gamma _n)$
coincides with
$\Lambda (\Gamma _\infty )$
.
The proof of the continuity statement, Theorem A(ii), is based on the following uniform equidistribution of the orbits. It is a quantified version of a result of Coornaert [Reference CoornaertCoo93].
Theorem B. Under the assumptions of Theorem A there exists
$K> 0$
such that, for every n and for every
$T\geq 0$
,
$$ \begin{align*}\frac{1}{K}\cdot e^{T\cdot h_{\Gamma_n}} \leq \Gamma_n x_n \cap \overline{B}(x_n,T) \leq K\cdot e^{T\cdot h_{\Gamma_n}}.\end{align*} $$
In the literature the behaviour of the critical exponent under another kind of convergence, algebraic convergence, was previously studied by Bishop and Jones in the case of hyperbolic manifolds [Reference Bishop and JonesBJ97] and in a more general setting by Paulin in [Reference PaulinPau97]. The definition of algebraic convergence, as well as the notation below, are recalled in §7. They proved the following result, which we refer to as the BJP theorem.
Theorem. [Reference Bishop and JonesBJ97, Reference PaulinPau97]
Let X be a geodesic,
$\delta $
-hyperbolic metric space such that for each
$x,y\in X$
there is a geodesic ray issuing from x passing at distance no more than
$\delta $
from y and let G be a finitely generated group. Let
$\varphi _n,\varphi _\infty \colon G \to \mathrm {Isom}(X)$
be homomorphisms. If
$\varphi _n(G)$
converges algebraically to
$\varphi _\infty (G)$
, if
$\varphi _n(G)$
,
$\varphi _\infty (G)$
are discrete and if
$\varphi _\infty (G)$
has no global fixed point at infinity, then
$h_{\varphi _\infty (G)} \leq \liminf _{n\to +\infty }h_{\varphi _n(G)}$
.
We point out here the main differences and analogies between this statement and Theorem A(ii).
-
• In the BJP theorem the isomorphism type of the group G is fixed. A posteriori this is not restrictive: under the assumptions of Theorem A the isomorphism type of the groups
$\Gamma _n$
is eventually constant (Corollary 7.7). The proof of Theorem A(i) does not use this property. -
• The algebraic limit is a priori different from the equivariant pointed Gromov–Hausdorff limit (see Example 7.3). However, if the spaces
$X_n$
are all isometric and satisfy the assumptions of Theorem A then the two limits coincide (Theorem 7.4). -
• In Theorem A the spaces
$X_n$
can be pairwise non-isometric. In that case the notion of algebraic convergence cannot be defined. This is a posteriori the main difference between algebraic convergence and equivariant pointed Gromov–Hausdorff convergence on the class
$\mathcal {M}(\delta ,D)$
.
2 Preliminaries on metric spaces
Throughout this paper X will denote a metric space and d will denote the metric on X. The open (respectively, closed) ball of radius r and centre x is denoted by
$B(x,r)$
(respectively,
$\overline {B}(x,r)$
). A geodesic segment is an isometry
$\gamma \colon I \to X$
where
$I=[a,b]$
is a bounded interval of
$\mathbb {R}$
. The points
$\gamma (a), \gamma (b)$
are called the endpoints of
$\gamma $
. A metric space X is said to be geodesic if for all couple of points
$x,y\in X$
there exists a geodesic segment whose endpoints are x and y. We will denote any geodesic segment between two points x and y, in an abuse of notation, by
$[x,y]$
. A geodesic ray is an isometry
$\gamma \colon [0,+\infty )\to X$
, while a geodesic line is an isometry
$\gamma \colon \mathbb {R}\to X$
.
The group of isometries of a proper metric space X (that is, closed balls are compact) is denoted by Isom
$(X)$
and it is endowed with the topology of uniform convergence on compact subsets of X.
If
$\Gamma $
is a subgroup of Isom
$(X)$
we define
$\Sigma _R(\Gamma , x) := \lbrace g\in \Gamma \text { s.t. } d(x,gx)\leq R\rbrace $
and
$\Gamma _R(x) := \langle \Sigma _R(\Gamma , x) \rangle $
, for every
$x\in X$
and
$R\geq 0$
. When the context is clear we simply write
$\Sigma _R(x)$
. A subgroup
$\Gamma $
is said to be discrete if equivalently:
-
(i) it is a discrete subspace of Isom
$(X)$
; -
(ii)
$\#\Sigma _R(x) < +\infty $
for all
$x\in X$
and all
$R\geq 0$
.
The systole of
$\Gamma $
at
$x\in X$
is the quantity
$$ \begin{align*}\text{sys}(\Gamma,x) := \inf\lbrace d(x,gx) \text{ s.t. } g\in \Gamma\setminus\lbrace \text{id}\rbrace\rbrace,\end{align*} $$
while the global systole of
$\Gamma $
is
$\text {sys}(\Gamma ,X) := \inf _{x\in X}\text {sys}(\Gamma ,x).$
If any non-trivial isometry of
$\Gamma $
has no fixed points then the systole at a point is always strictly positive by discreteness.
3 Convergence of group actions
First, we recall the definition and the properties of the equivariant pointed Gromov–Hausdorff convergence. Then we compare this notion with ultralimit convergence.
3.1 Equivariant pointed Gromov–Hausdorff convergence
We consider triples
$(X,x,\Gamma )$
where
$(X,x)$
is a pointed, proper metric space and
$\Gamma < \text {Isom}(X)$
. The following definitions are due to Fukaya [Reference FukayaFuk86].
Definition 3.1. Let
$(X,x,\Gamma ), (Y,y,\Lambda )$
be two triples as above and
$\varepsilon> 0$
. An equivariant
$\varepsilon $
-approximation between them is a triple
$(f,\varphi ,\psi )$
where:
-
•
$f \colon B(x, {1}/{\varepsilon }) \to B(y,{1}/{\varepsilon })$
is a map such that-
–
$f(x)=y$
, -
–
$\vert d(f(x_1), f(x_2)) - d(x_1, x_2)\vert <\varepsilon $
for every
$x_1,x_2\in B(x,{1}/{\varepsilon })$
, -
– for every
$y_1\in B(y, {1}/{\varepsilon })$
there exists
$x_1\in B(x,{1}/{\varepsilon })$
such that
$d(f(x_1),y_1) < \varepsilon $
;
-
-
•
$\varphi \colon \Sigma _{{1}/{\varepsilon }}(\Gamma , x) \to \Sigma _{{1}/{\varepsilon }}(\Lambda , y)$
is a map satisfying
$d(f(gx_1), \varphi (g) f(x_1)) <\varepsilon $
for every
$g\in \Sigma _{{1}/{\varepsilon }}(\Gamma , x)$
and every
$x_1\in B(x, {1}/{\varepsilon })$
such that also
$gx_1 \in B(x, {1}/{\varepsilon })$
; -
•
$\psi \colon \Sigma _{{1}/{\varepsilon }}(\Lambda , y) \to \Sigma _{{1}/{\varepsilon }}(\Gamma , x)$
is a map satisfying
$d(f(\psi (g)x_1), g f(x_1)) <\varepsilon $
for every
$g\in \Sigma _{{1}/{\varepsilon }}(\Lambda , y)$
and every
$x_1\in B(x, {1}/{\varepsilon })$
such that
$\psi (g)x_1 \in B(x, {1}/{\varepsilon })$
.
Definition 3.2. A sequence of triples
$(X_n,x_n,\Gamma _n)$
is said to converge in the equivariant pointed Gromov–Hausdorff sense to a triple
$(X,x,\Gamma )$
if for every
$\varepsilon> 0$
there exists
$n_\varepsilon \geq 0$
such that if
$n\geq n_\varepsilon $
then there exists an equivariant
$\varepsilon $
-approximation between
$(X_n,x_n,\Gamma _n)$
and
$(X,x,\Gamma )$
. One of these equivariant
$\varepsilon $
-approximations will be denoted by
$(f_n,\varphi _n,\psi _n)$
.
In this case we will write
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X,x,\Gamma )$
.
Remark 3.3. A few observations are in order.
-
• If
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X,x,\Gamma )$
then
$(X_n,x_n)$
converges in the classical pointed Gromov–Hausdorff sense to
$(X,x)$
. We denote this convergence by
$(X_n,x_n) \underset {\mathrm {pGH}}{\longrightarrow } (X,x)$
. -
• In the definition the limit space X is assumed to be proper. This is not restrictive, as we will see in a moment. If
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X,x,\Gamma )$
we denote by
$\hat {X}$
the completion of X. Any isometry of X defines uniquely an isometry of
$\hat {X}$
, so there is a well-defined group of isometries
$\hat {\Gamma }$
of
$\hat {X}$
associated to
$\Gamma $
. It follows from the definition that
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (\hat {X},\hat {x},\hat {\Gamma })$
too. Moreover, if a sequence of proper metric spaces converges in the pointed Gromov–Hausdorff sense to a complete metric space then the limit is proper by [Reference HerronHer16, Corollary 3.10].
We recall that Isom
$(X)$
is endowed with the topology of uniform convergence on compact subsets of X, when X is a proper space. It is classically known (see also the remark above) that the pointed Gromov–Hausdorff limit of a sequence of metric spaces is unique up to pointed isometry when we restrict to the class of complete (and therefore proper) spaces. In order to obtain uniqueness of the equivariant pointed Gromov–Hausdorff limit we need to restrict to groups that are closed in the isometry group of the limit space.
Proposition 3.4. [Reference FukayaFuk86, Proposition 1.5]
Suppose
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X,x,\Gamma )$
and
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (Y,y,\Lambda )$
, where
$X,Y$
are proper and
$\Gamma ,\Lambda $
are closed subgroups of Isom
$(X)$
, Isom
$(Y)$
, respectively. Then there exists an isometry
$F\colon X \to Y$
such that:
-
•
$F(x)=y$
; -
•
$F_*\colon \mathrm {Isom}(X) \to \mathrm {Isom}(Y)$
defined by
$F_*(g) = F\circ g \circ F^{-1}$
is an isomorphism between
$\Gamma $
and
$\Lambda $
.
Definition 3.5. Two triples
$(X,x,\Gamma )$
and
$(Y,y,\Lambda )$
are said equivariantly isometric if there exists an isometry
$F\colon X \to Y$
satisfying the thesis of the previous proposition. In this case F is called an equivariant isometry and we write
$(X,x,\Gamma )\cong (Y,y,\Lambda )$
.
From now on we will consider the equivariant pointed Gromov–Hausdorff convergence only restricted to triples
$(X,x,\Gamma )$
where
$(X,x)$
is a pointed proper metric space and
$\Gamma $
is a closed subgroup of Isom
$(X)$
. This condition is not restrictive.
Lemma 3.6. If
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X,x,\Gamma )$
then
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X,x,\bar \Gamma )$
, where
$\bar \Gamma $
is the closure of
$\Gamma $
in Isom
$(X)$
.
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 $
.
3.2 Ultralimit of groups
For more detailed notions on ultralimits we refer to [Reference Cavallucci and SambusettiCS21, Reference Druţu and KapovichDK18]. A non-principal ultrafilter
$\omega $
is a finitely additive measure on
$\mathbb {N}$
such that
$\omega (A) \in \lbrace 0,1 \rbrace $
for every
$A\subseteq \mathbb {N}$
and
$\omega (A)=0$
for every finite subset of
$\mathbb {N}$
. Accordingly we write
$\omega $
-a.s. and for
$\omega $
-a.e.
$(n)$
in the usual measure-theoretic sense.
Given a bounded sequence
$(a_n)$
of real numbers and a non-principal ultrafilter
$\omega $
, there exists a unique
$a\in \mathbb {R}$
such that for every
$\varepsilon> 0$
the set
$\lbrace n \in \mathbb {N} \text { s.t. } \vert a_n - a \vert < \varepsilon \rbrace $
has
$\omega $
-measure
$1$
; see, for instance, [Reference Druţu and KapovichDK18, Lemma 10.25]. The real number a is called the ultralimit of the sequence
$a_n$
and it is denoted by
$\omega $
-
$\lim a_n$
.
Given a sequence of pointed metric spaces
$(X_n, x_n)$
, we denote by
$(X_\omega , x_\omega )$
the ultralimit pointed metric space. It is the set of sequences
$(y_n)$
, where
$y_n\in X_n$
for every n, for which there exists
$M\in \mathbb {R}$
such that
$d(x_n,y_n)\leq M$
for
$\omega $
-a.e.
$(n)$
, modulo the relation
$(y_n)\sim (y_n')$
if and only if
$\omega $
-
$\lim d(y_n,y_n') = 0$
. The point of
$X_\omega $
defined by the class of the sequence
$(y_n)$
is denoted by
$y_\omega = \omega $
-
$\lim y_n$
. The formula
$d(\omega $
-
$\lim y_n, \omega $
-
$\lim y_n') = \omega $
-
$\lim d(y_n,y_n')$
defines a metric on
$X_\omega $
which is called the ultralimit distance on
$X_\omega $
.
A sequence of isometries
$g_n \in \text {Isom}(X_n)$
is admissible if there exists
$M\geq 0$
such that
$d(g_n x_n, x_n) \leq M \omega $
-a.s. Any such sequence defines an isometry
$g_\omega = \omega $
-
$\lim g_n$
of
$X_\omega $
by the formula
$g_\omega y_\omega = \omega $
-
$\lim g_ny_n$
[Reference Druţu and KapovichDK18, Lemma 10.48]. Given a sequence of groups of isometries
$\Gamma _n$
of
$X_n$
, we set
$$ \begin{align*}\Gamma_\omega = \lbrace \omega\text{-}\!\lim g_n \text{ s.t. } g_n \in \Gamma_n \text{ for } \omega\text{-a.e.}(n)\rbrace.\end{align*} $$
In particular, the elements of
$\Gamma _\omega $
are ultralimits of admissible sequences.
Lemma 3.7. The composition of admissible sequences of isometries is an admissible sequence of isometries and the limit of the composition is the composition of the limits.
(Indeed, if
$g_\omega = \omega $
-
$\lim g_n$
,
$h_\omega = \omega $
-
$\lim h_n$
belong to
$\Gamma _\omega $
then their composition belongs to
$\Gamma _\omega $
, as
$\omega $
-
$\lim d(g_n h_n \cdot x_n, x_n)\leq \omega $
-
$\lim d(g_n h_n \cdot x_n, g_n \cdot x_n) + \omega $
-
$\lim d(g_n \cdot x_n, x_n) < + \infty $
.)
Analogously one proves that
$(\text {id}_n)$
belongs to
$\Gamma _\omega $
and defines the identity map of
$X_\omega $
, and that if
$g_\omega = \omega $
-
$\lim g_n$
belongs to
$\Gamma _\omega $
then also the sequence
$(g_n^{-1})$
defines an element of
$\Gamma _\omega $
, which is the inverse of
$g_\omega $
.
So we have a well-defined composition law on
$\Gamma _\omega $
, that is, for
$ g_\omega = \omega $
-
$\lim g_n$
and
$ h_\omega = \omega $
-
$\lim h_n$
we set
$g_\omega \, \circ \, h_\omega = \omega $
-
$\lim (g_n \circ h_n).$
With this operation
$\Gamma _\omega $
is a group of isometries of
$X_\omega $
and we call it the ultralimit group of the sequence of groups
$\Gamma _n$
.
The ultralimit space
$X_\omega $
may be not proper in general, even if
$X_n$
is proper for every n. When
$X_\omega $
is proper,
$\Gamma _\omega $
is closed with respect to the uniform convergence on compact subsets.
Proposition 3.8. Let
$(X_n,x_n,\Gamma _n)$
be a sequence of proper metric spaces and
$\omega $
be a non-principal ultrafilter. If
$X_\omega $
is proper then
$\Gamma _\omega $
is a closed subgroup of Isom
$(X_\omega )$
.
We remark that the proof is analogous to [Reference Druţu and KapovichDK18, Corollary 10.64].
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 $
.
Lemma 3.9. Let
$(X,x)$
be a proper metric space and
$\Gamma \subseteq \mathrm {Isom}(X)$
be a closed subgroup. Then the ultralimit of the constant sequence
$(X,x,\Gamma )$
is naturally equivariantly isometric to
$(X,x,\Gamma )$
for every non-principal ultrafilter.
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 $
.
3.3 Comparison between the two convergences
We now compare the ultralimit convergence to the equivariant pointed Gromov–Hausdorff convergence. Analogues of the next results for the classical pointed Gromov–Hausdorff convergence can be found, for instance, in [Reference JansenJan17].
Proposition 3.10. Suppose
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X,x,\Gamma )$
and denote by
$(f_n,\varphi _n,\psi _n)$
some corresponding equivariant approximations. Let
$\omega $
be a non-principal ultrafilter and let
$(X_\omega , x_\omega , \Gamma _\omega )$
be the ultralimit triple. Then the map
$$ \begin{align*}F \colon (X_\omega, x_\omega, \Gamma_\omega) \to (X,x,\Gamma)\end{align*} $$
defined by sending
$y_\omega = \omega $
-
$\lim y_n \in X_\omega $
to
$\iota ^{-1}(\omega $
-
$\lim f_n(y_n))$
is a well-defined equivariant isometry. Here
$\iota $
is the natural equivariant isometry of Lemma 3.9 between
$(X,x,\Gamma )$
and the ultralimit of its constant sequence.
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 $
.
Proposition 3.11. Let
$(X_n,x_n,\Gamma _n)$
be a sequence of triples,
$\omega $
be a non-principal ultrafilter and
$(X_\omega , x_\omega , \Gamma _\omega )$
be the ultralimit triple. If
$X_\omega $
is proper then there exists a subsequence
$\lbrace n_k \rbrace \subseteq \mathbb {N}$
such that
$(X_{n_k},x_{n_k},\Gamma _{n_k}) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X_\omega ,x_\omega ,\Gamma _\omega )$
.
Remark 3.12. Notice that the subsequence
$\lbrace n_k \rbrace $
may not belong to
$\omega $
.
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.
We summarize these properties in the following proposition.
Proposition 3.13. Let
$(X_n, x_n, \Gamma _n)$
be a sequence of triples and let
$\omega $
be a non-principal ultrafilter.
-
(i) If
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X,x,\Gamma )$
then
$(X_\omega , x_\omega , \Gamma _\omega ) \cong (X,x,\Gamma )$
. -
(ii) If
$X_\omega $
is proper then
$(X_{n_k},x_{n_k},\Gamma _{n_k}) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X_\omega ,x_\omega ,\Gamma _\omega )$
for some subsequence
$\lbrace {n_k}\rbrace $
.
Corollary 3.14. Let
$(X_n, x_n, \Gamma _n)$
be a sequence of triples and suppose that there is a triple
$(X,x,\Gamma )$
, X proper, such that
$(X,x,\Gamma ) \cong (X_\omega ,x_\omega ,\Gamma _\omega )$
for every non-principal ultrafilter
$\omega $
. Then
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X,x,\Gamma )$
.
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
$(X,x,\Gamma )$
.
4 Gromov hyperbolic metric spaces
We recall briefly the definition and some properties of Gromov-hyperbolic metric spaces. Good references are, for instance, [Reference Bridson and HaefligerBH13, Reference Coornaert, Delzant and PapadopoulosCDP90].
Let X be a geodesic metric space. Given three points
$x,y,z \in X$
, the Gromov product of y and z with respect to x is defined as
$$ \begin{align*}(y,z)_x = \tfrac{1}{2}( d(x,y) + d(x,z) - d(y,z) ).\end{align*} $$
The space X is said to be
$\delta $
-hyperbolic,
$\delta \geq 0$
, if for every four points
$x,y,z,w \in X$
the following four-points condition holds:
$$ \begin{align} (x,z)_w \geq \min\lbrace (x,y)_w, (y,z)_w \rbrace - \delta \end{align} $$
or, equivalently,
$$ \begin{align} d(x,y) + d(z,w) \leq \max \lbrace d(x,z) + d(y,w), d(x,w) + d(y,z) \rbrace + 2\delta. \end{align} $$
The space X is Gromov hyperbolic if it is
$\delta $
-hyperbolic for some
$\delta \geq 0$
.
This formulation of
$\delta $
-hyperbolicity is convenient when one is interested in taking limits. We will also make use of another classical characterization of
$\delta $
-hyperbolicity. A geodesic triangle in X is the union of three geodesic segments
$[x,y], [y,z], [z,x]$
and is denoted by
$\Delta (x,y,z)$
. For every geodesic triangle there exists a unique tripod
$\overline \Delta $
with vertices
$\bar {x},\bar {y},\bar {z}$
such that the lengths of
$[\bar {x}, \bar {y}], [\bar {y}, \bar {z}], [\bar {z}, \bar {x}]$
equal the lengths of
$[x,y], [y,z], [z,x]$
, respectively. There exists a unique map
$f_\Delta $
from
$\Delta (x,y,z)$
to the tripod
$\overline \Delta $
that isometrically identifies the corresponding edges, and there are exactly three points
$c_x \in [y,z], c_y \in [x,z], c_z\in [x,y]$
such that
$f_\Delta (c_x) = f_\Delta (c_y) = f_\Delta (c_z) = c$
, where c is the centre of the tripod
$\overline \Delta $
. By definition of
$f_\Delta $
, we have
$$ \begin{align*}d(x,c_z) = d(x,c_y), \quad d(y,c_x) = d(y,c_z), \quad d(z,c_x)=d(z,c_y).\end{align*} $$
The triangle
$\Delta (x,y,z)$
is called
$\delta $
-thin if for every
$u,v \in \Delta (x,y,z)$
such that
$f_\Delta (u)=f_\Delta (v)$
the inequality
$d(u,v)\leq \delta $
holds; in particular, the mutual distances between
$c_x,c_y$
and
$c_z$
are at most
$\delta $
. It is well known that every geodesic triangle in a geodesic
$\delta $
-hyperbolic metric space (as defined above) is
$4\delta $
-thin.
Moreover, the last condition is equivalent to the above definition of hyperbolicity, up to slightly increasing the hyperbolicity constant
$\delta $
in (4.1).
The following is a basic property of Gromov-hyperbolic metric spaces.
Lemma 4.1. (Projection lemma, cf. [Reference Coornaert, Delzant and PapadopoulosCDP90, Lemma 3.2.7])
Let X be a
$\delta $
-hyperbolic metric space and let
$x,y,z \in X$
. For every geodesic segment
$[y,z]$
we have
$(y,z)_x \geq d(x, [y,z]) - 4\delta .$
Let X be a proper,
$\delta $
-hyperbolic metric space and x be a point of X. The Gromov boundary of X is defined as the quotient
$$ \begin{align*}\partial X = \lbrace (z_n)_{n \in \mathbb{N}} \subseteq X \mid \lim_{n,m \to +\infty} (z_n,z_m)_{x} = + \infty \rbrace /_\approx,\end{align*} $$
where
$(z_n)_{n \in \mathbb {N}}$
is a sequence of points in X and
$\approx $
is the equivalence relation defined by
$(z_n)_{n \in \mathbb {N}} \approx (z_n')_{n \in \mathbb {N}}$
if and only if
$\lim _{n,m \to +\infty } (z_n,z_m')_{x} = + \infty $
. We will write
$ z = [(z_n)] \in \partial X$
for short, and we say that
$(z_n)$
converges to z. This definition does not depend on the basepoint x.
There is a natural topology on
$X\cup \partial X$
that extends the metric topology of X. The Gromov product can be extended to points
$z,z' \in \partial X$
by
$$ \begin{align*}(z,z')_{x} = \sup_{(z_n) , (z_n') } \liminf_{n,m \to + \infty} (z_n, z_m')_{x}\end{align*} $$
where the supremum is taken among all sequences such that
$(z_n) \in z$
and
$(z_n')\in z'$
. For every
$z,z',z'' \in \partial X$
the inequality
$$ \begin{align} (z,z')_{x} \geq \min\lbrace (z,z'')_{x}, (z',z'')_{x} \rbrace - \delta. \end{align} $$
continues to hold. Moreover, for all sequences
$(z_n),(z_n')$
converging to
$z,z'$
respectively, we have
$$ \begin{align} (z,z')_{x} -\delta \leq \liminf_{n,m \to + \infty} (z_n,z_m')_{x} \leq (z,z')_{x}. \end{align} $$
The Gromov product between a point
$y\in X$
and a point
$z\in \partial X$
is defined in a similar way and satisfies a condition analogous to (4.4).
Every geodesic ray
$\xi $
defines a point
$\xi ^+=[(\xi (n))_{n \in \mathbb {N}}]$
of the Gromov boundary
$ \partial X$
: we say that
$\xi $
joins
$\xi (0) = y$
to
$\xi ^+ = z$
, and we denote this by
$[y, z]$
. Moreover, for every
$z\in \partial X$
and every
$x\in X$
it is possible to find a geodesic ray
$\xi $
such that
$\xi (0)=x$
and
$\xi ^+ = z$
. Any such geodesic ray is denoted by
$\xi _{x,z} = [x,z]$
even if it is possibly not unique. Analogously, given different points
$z = [(z_n)], z' = [(z^{\prime }_n)] \in \partial X$
, there always exists a geodesic line
$\gamma $
joining
$z'$
to z, that is, such that
$\gamma |_{[0, +\infty )}$
and
$\gamma |_{(-\infty ,0]}$
join
$\gamma (0)$
to
$z,z'$
, respectively (just consider the limit
$\gamma $
of the segments
$[z_n,z^{\prime }_n]$
; notice that all these segments intersect a ball of fixed radius centred at
$x_0$
, since
$(z_n,z^{\prime }_m)_{x_0}$
is uniformly bounded above). We call z and
$z'$
respectively the positive and negative endpoints of
$\gamma $
, denoted by
$\gamma ^\pm $
. The relation between Gromov product and geodesic ray is highlighted in the following well-known lemma.
Lemma 4.2. Let X be a proper,
$\delta $
-hyperbolic metric space,
$z,z'\in \partial X$
and
$x\in X$
.
-
(i) If
$(z,z')_{x} \geq T$
then
$d(\xi _{x,z}(T - \delta ),\xi _{x,z'}(T - \delta )) \leq 4\delta $
. -
(ii) If
$d(\xi _{x,z}(T),\xi _{x,z'}(T)) < 2b$
then
$(z,z')_{x}> T - b$
, for all
$b>0$
.
Proof. Assume
$(z,z')_{x} \geq T$
and suppose
$d(\xi _{x,z}(T - \delta ),\xi _{x,z'}(T - \delta ))> 4\delta $
. Fix
$S\geq T - \delta $
and consider the triangle
$\Delta (x, \xi _{x,z}(S), \xi _{x,z'}(S))$
. There exist
$a\in [x,\xi _{x,z}(S)], b\in [x,\xi _{x,z'}(S)], c\in [\xi _{x,z}(S), \xi _{x,z'}(S)]$
such that
$d(a,b)<\delta ,\, d(b,c)<\delta ,\, d(a,c)<\delta $
and
$T_\delta \,:=\, d(x,a)=d(x,b), d(\xi _{x,z}(S),a) \,=\, d(\xi _{x,z}(S),c),\,\, d(\xi _{x,z'}(S),b) = d(\xi _{x,z'}(S), c)$
. Since this triangle is
$4\delta $
-thin we conclude that
$T \kern-1pt - \delta \kern-1pt >\kern-1pt T_\delta $
. Moreover,
$d(\xi _{x,z}(S),\xi _{x,z'} (S)) = d(\xi _{x,z}(S),c) + d(c,\xi _{x,z'}(S)) = 2(S - T_\delta ).$
Hence
$$ \begin{align*} \begin{aligned} (z,z')_{x} \leq \liminf_{S\to + \infty}\tfrac{1}{2}( 2S - d(\xi_{x,z}(S),\xi_{x,z'}(S)) ) + \delta = T_\delta + \delta < T, \end{aligned} \end{align*} $$
where we have used (4.4). This contradiction concludes (i).
Now we assume
$d(\xi _{x,z}(T),\xi _{x,z'}(T)) < 2b$
. Applying (4.4) again and using
$d(\xi _{x,z}(S), \xi _{x,z'}(S)) < 2(S-T) + 2b$
for all
$S\geq T$
, we obtain
$$ \begin{align*}(z,z')_{x} \geq \liminf_{S\to + \infty}\tfrac{1}{2}( 2S - d(\xi_{x,z}(S),\xi_{x,z'}(S)) )> T + b.\\[-3.5pc]\end{align*} $$
Remark 4.3. We remark that the computation above shows also that if
$z\in \partial X$
,
$y\in X$
and
$(y,z)_x \geq T$
then
$d(x,y)> T-\delta $
and
$d(\gamma (T-\delta ), \xi _{x,z}(T-\delta )) \leq 4\delta $
for every geodesic segment
$\gamma = [x,y]$
.
The following is a standard computation; see, for instance, [Reference Besson, Courtois, Gallot and SambusettiBCGS17].
Lemma 4.4. Let X be a proper,
$\delta $
-hyperbolic metric space. Then every pair of geodesic rays
$\xi , \xi '$
with same endpoints at infinity are at distance at most
$8\delta $
, that is, there exist
$t_1,t_2\geq 0$
such that
$t_1+t_2=d(\xi (0),\xi '(0))$
and
$d(\xi (t + t_1),\xi '(t+t_2)) \leq 8\delta $
for all
$t\in \mathbb {R}$
.
A curve
$\alpha \colon [a,b] \to X$
is a
$(1,\nu )$
-quasigeodesic,
$\nu \geq 0$
, if
$$ \begin{align*}\vert s - t \vert - \nu \leq d(\alpha(s),\alpha(t)) \leq \vert s - t \vert + \nu\end{align*} $$
for all
$s,t\in [a,b]$
. A subset Y of X is said to be
$\unicode{x3bb} $
-quasiconvex if every point of every geodesic segment joining every pair of points
$y,y'$
of Y is at distance at most
$\unicode{x3bb} $
from Y. The quasiconvex hull of a subset C of
$\partial X$
is the union of all the geodesic lines joining two points of C and is denoted by QC-Hull
$(C)$
. The following lemma justifies this name.
Lemma 4.5. Let X be a proper,
$\delta $
-hyperbolic metric space and let C be a subset of
$\partial X$
. Then QC-Hull
$(C)$
is
$36\delta $
-quasiconvex. Moreover, if C is closed then QC-Hull
$(C)$
is closed.
Proof. Let
$x,y\in \textrm{QC-Hull}(C)$
. By definition they belong to geodesics
$\gamma _x,\gamma _y$
with both endpoints in C. We parametrize
$\gamma _x$
and
$\gamma _y$
in such a way that
$d(\gamma _x(0),\gamma _y(0)) = d(\gamma _x,\gamma _y)$
and
$x=\gamma _x(t_x)$
,
$y=\gamma _y(t_y)$
with
$t_x,t_y \geq 0$
. We take a geodesic
$\gamma = [\gamma _x^+, \gamma _y^+] \subseteq \textrm{QC-Hull}(C)$
. By Lemma 4.4 there are points
$x',y'\in \gamma $
at distance at most
$8\delta $
from x and y, respectively. Therefore the path
$\alpha = [x,x']\cup [x',y'] \cup [y',y]$
is a
$(1,16\delta )$
-quasigeodesic. By a standard computation in hyperbolic geometry (see, for instance, [Reference Cavallucci and SambusettiCS20, Proposition 3.5(a)]) we conclude that any point of
$[x,y]$
is at distance at most
$28\delta $
from a point of
$\alpha $
and so at distance at most
$36\delta $
from a point of
$\gamma $
. This concludes the proof of the first part since the points of
$\gamma $
are in the quasiconvex hull of C.
Suppose now that there are points
$x_n\in \textrm{QC-Hull}(C)$
converging to
$x_\infty \in X$
. By definition
$x_n \in \gamma _n$
, where
$\gamma _n$
is a geodesic line with endpoints
$\gamma _n^\pm \in C$
. The geodesics
$\gamma _n$
converge uniformly on compact subsets to a geodesic
$\gamma _\infty $
containing
$x_\infty $
, since X is proper. The sequences
$\gamma _n^\pm $
converge to the endpoints of
$\gamma _\infty $
(cf. [Reference Bartels and LückBL12, Lemma 1.6]). Using the fact that C is closed, we conclude that
$\gamma _\infty ^\pm \in C$
, that is,
$x_\infty \in \textrm{QC-Hull}(C)$
.
We need the following approximation result.
Lemma 4.6. Let X be a proper,
$\delta $
-hyperbolic metric space. Let
$C\subseteq \partial X$
be a subset with at least two points and
$x\in \mathrm {QC-Hull}(C)$
. Then for every
$z\in C$
there exists a geodesic line
$\gamma $
with endpoints in C such that
$d(\xi _{x,z}(t), \gamma (t)) \leq 14\delta $
for every
$t\geq 0$
. In particular,
$d(\xi _{x,z}(t), \mathrm {QC-Hull}(C)) \leq 14\delta $
.
Proof. Since
$x\in \mathrm {QC-Hull}(C))$
, there exists a geodesic line
$\eta $
joining two points
$\eta ^\pm $
of C such that
$x\in \eta $
. Of course we have
$(\eta ^+,\eta ^-)_x \leq \delta $
, so by (4.3) we get
$$ \begin{align*}\delta \geq (\eta^+,\eta^-)_x \geq \min\lbrace (\eta^+,z)_x, (\eta^-,z)_x\rbrace - \delta.\end{align*} $$
Therefore one of the two values
$(\eta ^+,z)_x$
,
$(\eta ^-,z)_x$
is less than or equal to
$2\delta $
. Let us suppose it is the former. We consider a geodesic line
$\gamma $
joining
$\eta ^+$
and z. By Lemma 4.1 we get
$$ \begin{align*}d(x,\gamma([-S,S])) \leq (\gamma(-S), \gamma(S))_x + 4\delta\end{align*} $$
for every
$S\geq 0$
. Taking
$S\to +\infty $
, the points
$\gamma (-S)$
and
$\gamma (S)$
converge respectively to
$\eta ^+$
and z. Therefore by (4.4) we get
$d(x,\gamma ) \leq 6\delta $
. If we parametrize
$\gamma $
so that
$d(x,\gamma (0)) \leq 6\delta $
then
$d(\xi _{x,z}(t), \gamma (t)) \leq 14\delta $
for every
$t\geq 0$
, by Lemma 4.4.
The Busemann function associated to
$z\in \partial X$
with basepoint x is
$$ \begin{align*}B_z(x,\cdot)\colon X \to \mathbb{R},\quad y \mapsto \lim_{T\to +\infty} (d(\xi_{x,z}(T), y) - T).\end{align*} $$
It depends on the choice of the geodesic ray
$\xi _{x,z}=[x,z]$
, but two maps obtained by taking two different geodesic rays are at bounded distance and the bound depends only on
$\delta $
. Every Busemann function is
$1$
-Lipschitz.
4.1 Visual metrics
When X is a proper,
$\delta $
-hyperbolic metric space it is known that the boundary
$\partial X$
is metrizable. A metric
$D_{x,a}$
on
$\partial X$
is called a visual metric of centre
$x \in X$
and parameter
$a\in (0,{1}/({2\delta \cdot \log _2e}))$
if there exists
$V> 0$
such that for all
$z,z' \in \partial X$
the inequality
$$ \begin{align} \frac{1}{V}e^{-a(z,z')_{x}}\leq D_{x,a}(z,z')\leq V e^{-a(z,z')_{x}}. \end{align} $$
holds. A visual metric is said to be standard if for all
$z,z'\in \partial X$
, we have
$$ \begin{align*}(3-2e^{a\delta})e^{-a(z,z')_{x}}\leq D_{x,a}(z,z')\leq e^{-a(z,z')_{x}}.\end{align*} $$
For all a as before and
$x\in X$
there exists always a standard visual metric of centre x and parameter a; see [Reference PaulinPau96]. The generalized visual ball of centre
$z \in \partial X$
and radius
$\rho \geq 0$
is
$$ \begin{align*}B(z,\rho) = \bigg\lbrace z' \in \partial X \text{ s.t. } (z,z')_{x}> \log \frac{1}{\rho} \bigg\rbrace.\end{align*} $$
It is comparable to the metric balls of the visual metrics on
$\partial X$
.
Lemma 4.7. Let
$D_{x,a}$
be a visual metric of centre x and parameter a on
$\partial X$
. Then for all
$z\in \partial X$
and for all
$\rho>0$
we have
$$ \begin{align*}B_{D_{x,a}}\bigg(z, \frac{1}{V}\rho^a\bigg) \subseteq B(z,\rho)\subseteq B_{D_{x,a}}(z, V\rho^a ).\end{align*} $$
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,
$z'\in B(z,\rho )$
.
It is classical that generalized visual balls are related to shadows, whose definition is as follows. The shadow of radius
$r>0$
cast by a point
$y\in X$
with centre
$x\in X$
is the set
$$ \begin{align*}\text{Shad}_x(y,r) = \lbrace z\in \partial X \text{ s.t. } [x,z]\cap B(y,r) \neq \emptyset \text{ for all rays } [x,z]\rbrace.\end{align*} $$
Lemma 4.8. Let X be a proper,
$\delta $
-hyperbolic metric space. Let
$z\in \partial X$
,
$x\in X$
and
$T\geq 0$
. Then:
-
(i)
$B(z,e^{-T}) \subseteq \mathrm {Shad}_{x}(\xi _{x,z}(T), 7\delta )$
; -
(ii)
$\mathrm {Shad}_{x}(\xi _{x,z}(T), r) \subseteq B(z, e^{-T + r})$
for all
$r> 0$
.
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).
A compact metric space Z is
$(A,s)$
-Ahlfors regular if there exists a probability measure
$\mu $
on Z such that
$$ \begin{align*}\frac{1}{A}\rho^s \leq \mu(B(z,\rho)) \leq A\rho^s\end{align*} $$
for all
$z\in Z$
and all
$0\leq \rho \leq \text {Diam}(Z)$
, where Diam
$(Z)$
is the diameter of Z. If
$Z=\partial X$
we say that Z is visually
$(A,s)$
-Ahlfors regular if there exists a probability measure
$\mu $
on
$\partial X$
such that
$$ \begin{align*}\frac{1}{A}\rho^s \leq \mu(B(z,\rho)) \leq A\rho^s\end{align*} $$
for all
$z\in Z$
and all
$0\leq \rho \leq 1$
, where
$B(z,\rho )$
is the generalized visual ball of centre z and radius
$\rho $
. The following result is an immediate consequence of Lemma 4.7.
Lemma 4.9. If
$\partial X$
is
$(A,s)$
-Ahlfors regular with respect to a visual metric of centre x and parameter a, then it is visually
$(AV^s,as)$
-Ahlfors regular, where V is the constant of (4.5).
The packing
$^*$
number at scale
$\rho $
of a subset C of the boundary of a proper Gromov-hyperbolic space
$\partial X$
is the maximal number of disjoint generalized visual balls of radius
$\rho $
with centre in C. We denote it by
$\text {Pack}^*(C, \rho )$
. We write
$\text {Cov}(C, \rho )$
to denote the minimal number of generalized visual balls of radius
$\rho $
needed to cover C.
Lemma 4.10. For all
$T\geq 0$
we have the inequalities
$\mathrm {Pack}^*(C, e^{-T +\delta }) \leq \mathrm {Cov}(C, e^{-T})$
and
$\mathrm {Cov}(C, e^{-T +\delta }) \leq \mathrm {Pack}^*(C, e^{-T}).$
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.
4.2 Groups of isometries, limit set and critical exponent
Let X be a proper,
$\delta $
-hyperbolic metric space. Every isometry of X acts naturally on
$\partial X$
and the resulting map on
$X\cup \partial X$
is a homeomorphism. The limit set
$\Lambda (\Gamma )$
of a discrete group of isometries
$\Gamma $
is the set of accumulation points of the orbit
$\Gamma x$
on
$\partial X$
, where x is any point of X. It is the smallest
$\Gamma $
-invariant closed set of the Gromov boundary, indeed we have the following proposition.
Proposition 4.11. [Reference CoornaertCoo93, Theorem 5.1]
Let
$\Gamma $
be a discrete group of isometries of a proper, Gromov-hyperbolic metric space. Then
$\Lambda (\Gamma )$
is the smallest closed
$\Gamma $
-invariant subset of
$\partial X$
, that is, every
$\Gamma $
-invariant, closed subset C of
$\partial X$
contains
$\Lambda (\Gamma )$
.
The group
$\Gamma $
is called elementary if
$\# \Lambda (\Gamma ) \leq 2$
. The limit superior in the definition of the critical exponent of
$\Gamma $
is a true limit.
Lemma 4.12. [Reference CavallucciCav21b, Theorem B]
Let X be a proper,
$\delta $
-hyperbolic metric space and let
$\Gamma $
be a discrete group of isometries of X. Then
$$ \begin{align*}h_\Gamma = \lim_{T\to + \infty} \frac{1}{T} \log \# \Gamma x \cap \overline{B}(x,T).\end{align*} $$
The critical exponent of
$\Gamma $
can be seen also as
$$ \begin{align*}h_\Gamma = \inf\bigg\lbrace s \geq 0 \text{ s.t. } \sum_{g \in \Gamma}e^{-sd(x,g x)} < +\infty\bigg\rbrace.\end{align*} $$
We remark that for every
$s \geq 0$
the series
$\sum _{g \in \Gamma }e^{-sd(x,g x)}$
, which is called the Poincaré series of
$\Gamma $
, is
$\Gamma $
-invariant. In other words,
$\sum _{g \in \Gamma }e^{-sd(x,g x)} = \sum _{g \in \Gamma }e^{-sd(x',g x')}$
for all
$x' \in \Gamma x$
. There is a canonical way to construct a measure on
$\partial X$
starting from the Poincaré series. For every
$s>h_\Gamma $
the measure
$$ \begin{align*}\mu_s = \frac{1}{\sum_{g \in \Gamma}e^{-sd(x,g x)}}\cdot \sum_{g \in \Gamma}e^{-sd(x,g x)}\Delta_{g x},\end{align*} $$
where
$\Delta _{g x}$
is the Dirac measure at
$g x$
, is a probability measure on the compact space
$X\cup \partial X$
. Then there exists a sequence
$s_i$
converging to
$h_\Gamma $
such that
$\mu _{s_i}$
converges
$*$
-weakly to a probability measure on
$X\cup \partial X$
. Any of these limits is called a Patterson–Sullivan measure and is denoted by
$\mu _{\text {PS}}$
.
Proposition 4.13. [Reference CoornaertCoo93, Theorem 5.4]
Let X be a proper,
$\delta $
-hyperbolic metric space and let
$\Gamma $
be a discrete group of isometries of X with
$h_\Gamma <+\infty $
. Then every Patterson–Sullivan measure is supported on
$\Lambda (\Gamma )$
. Moreover, it is a
$\Gamma $
-quasiconformal density of dimension
$h_\Gamma $
, that is, it satisfies
$$ \begin{align*}\frac{1}{Q}\cdot e^{h_\Gamma (B_z(x,x) - B_z(x,gx))} \leq \frac{d(g_* \mu_{\mathrm{PS}})}{d\mu_{\mathrm{PS}}}(z) \leq Q\cdot e^{h_\Gamma (B_z(x,x) - B_z(x,gx))}\end{align*} $$
for every
$g \in \Gamma $
and every
$z\in \Lambda (\Gamma )$
, where Q is a constant depending only on
$\delta $
and an upper bound on
$h_\Gamma $
.
The quantification of Q is not explicated in the original paper, but it follows from the proof therein.
The set
$\Lambda (\Gamma )$
is
$\Gamma $
-invariant so it is its quasiconvex hull. We recall that a discrete group of isometries
$\Gamma $
is quasiconvex-cocompact if and only if its action on QC-Hull
$(\Lambda (\Gamma ))$
is cocompact, that is, if there exists
$D\geq 0$
such that for all
$x,y\in \textrm{QC-Hull}(\Lambda (\Gamma ))$
the inequality
$d(gx,y)\leq D$
holds for some
$g\in \Gamma $
. The smallest D satisfying this property is called the codiameter of
$\Gamma $
.
Given two real numbers
$\delta \geq 0$
and
$D>0$
, we recall that
$\mathcal {M}(\delta ,D)$
is the class of triples
$(X,x,\Gamma )$
, where X is a proper, geodesic,
$\delta $
-hyperbolic metric space,
$\Gamma $
is a discrete, non-elementary, torsion-free, quasiconvex-cocompact group of isometries with codiameter
$\leq D$
and
$x\in \mathrm {QC-Hull}(\Lambda (\Gamma ))$
. For an element
$(X,x,\Gamma )$
of
$\mathcal {M}(\delta ,D)$
we will use Y to denote
$\mathrm {QC-Hull}(\Lambda (\Gamma ))$
.
5
$\mathcal {M}(\delta ,D)$
is closed under equivariant Gromov–Hausdorff limits
The purpose of this section is to prove statement (i) of Theorem A. We need to understand better the properties of the spaces belonging to
$\mathcal {M}(\delta ,D)$
.
5.1 Entropy and systolic estimates on
$\mathcal {M}(\delta ,D)$
The following are straightforward adaptations of results of [Reference Besson, Courtois, Gallot and SambusettiBCGS17, Reference Besson, Courtois, Gallot and SambusettiBCGS21].
Lemma 5.1. If
$(X,x,\Gamma ) \in \mathcal {M}(\delta ,D)$
then
$\Sigma _{2D+72\delta }(x)$
generates
$\Gamma $
.
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
$g_i \in \Sigma _{2D+72\delta }(x)$
.
Proposition 5.2. [Reference Besson, Courtois, Gallot and SambusettiBCGS17, Proposition 5.10]
If
$(X,x,\Gamma ) \in \mathcal {M}(\delta ,D)$
then
$h_\Gamma \geq {\log 2}/({99\delta + 10D})$
.
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.
Proposition 5.3. [Reference Besson, Courtois, Gallot and SambusettiBCGS21, Theorem 3.4]
For every
$H \geq 0$
there exists
$s=s(\delta ,D,H)> 0$
such that if
$(X,x,\Gamma ) \in \mathcal {M}(\delta ,D)$
and if
$h_\Gamma \leq H$
then sys
$(\Gamma ,X) \geq s.$
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.
5.2 Covering entropy
Let Y be any subset of a metric space X.
-
• A subset S of Y is called r-dense if for all
$y \in Y$
there exists
$z\in S$
such that
$d(y,z)\leq r$
. -
• A subset S of Y is called r-separated if
$d(y,z)> r$
for all
$y,z \in S$
.
The packing number of Y at scale r is the maximal cardinality of a
$2r$
-separated subset of Y and is denoted by
$\text {Pack}(Y,r)$
. The covering number of Y is the minimal cardinality of an r-dense subset of Y and is denoted by
$\text {Cov}(Y,r)$
. These two quantities are classically related by
$$ \begin{align} \text{Pack}(Y,2r) \leq \text{Cov}(Y,2r) \leq \text{Pack}(Y,r). \end{align} $$
Y is said to be uniformly packed at scales
$0<r\leq R$
if
$$ \begin{align*}\text{Pack}_Y(R,r) := \sup_{x\in Y}\text{Pack}(\overline{B}(x,R) \cap Y,r) <+\infty\end{align*} $$
and uniformly covered at scales
$0<r\leq R$
if
$$ \begin{align*}\text{Cov}_Y(R,r):=\sup_{x\in Y}\text{Cov}(\overline{B}(x,R) \cap Y,r) <+\infty.\end{align*} $$
Lemma 5.4. Let
$(X,x,\Gamma )\in \mathcal {M}(\delta ,D)$
. Then
$Y=\mathrm {QC-Hull}(\Lambda (\Gamma ))$
is uniformly packed and uniformly covered at any scales.
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.
We recall the notion of covering entropy. This was studied by the author in a less general context in [Reference CavallucciCav21a].
Definition 5.5. Let X be a proper metric space and
$x\in X$
. The upper covering entropy of X at scale
$r>0$
is the quantity
$$ \begin{align*}\overline{h}_{\mathrm{Cov}}(X,r) = \limsup_{T\to + \infty}\frac{\log \mathrm{Cov}(\overline{B}(x,T),r)}{T},\end{align*} $$
while the lower covering entropy of X at scale r is
$$ \begin{align*}\underline{h}_{\mathrm{Cov}}(X,r) = \liminf_{T\to + \infty}\frac{\log \mathrm{Cov}(\overline{B}(x,T),r)}{T}.\end{align*} $$
They do not depend on the point
$x\in X$
by a standard argument.
Lemma 5.6. Let
$(X,x,\Gamma )\in \mathcal {M}(\delta ,D)$
. If there exist
$r,P>0$
such that Pack
$_{Y}(72\delta + 3r, r) \leq P$
then Pack
$_{Y}(T, r) \leq P\cdot (1+P)^{({T}/{r}) - 1}$
for every
$T\geq 0$
. In particular,
$\overline {h}_{\mathrm {Cov}}(Y,2r) \leq ({\log (1+P)})/{r}$
.
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).
Proposition 5.7. Let
$(X,x,\Gamma )\in \mathcal {M}(\delta ,D)$
. Then
$\overline {h}_{\mathrm {Cov}}(Y,r)$
and
$\underline {h}_{\mathrm {Cov}}(Y,r)$
do not depend on r and the same quantities can be defined by replacing the covering function with the packing function. Moreover, they coincide and
$$ \begin{align*}h_{\mathrm{Cov}}(Y):=\lim_{T\to + \infty}\frac{\log \mathrm{Cov}(\overline{B}(x,T)\cap Y,r)}{T} = h_\Gamma < +\infty.\end{align*} $$
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.
5.3 Convergence of spaces in
$\mathcal {M}(\delta ,D)$
The following situation will be called the standard setting of convergence: we have a sequence
$(X_n,x_n,\Gamma _n) \in \mathcal {M}(\delta ,D)$
such that
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X_\infty , x_\infty ,\Gamma _\infty )$
. Observe that
$X_\infty $
is a proper metric space by definition.
Lemma 5.8. In the standard setting of convergence
$\sup _{n\in \mathbb {N}}\mathrm {Pack}_{Y_n}(R,r)< +\infty $
for every
$0<r\leq R$
.
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
$Y_n$
.
Corollary 5.9. In the standard setting of convergence
$\sup _{n\in \mathbb {N}}h_{\Gamma _n} < +\infty $
.
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
$$ \begin{align*}h_{\Gamma_n} = h_{\mathrm{Cov}}(Y_n) \leq \log(1+P).\\[-3pc]\end{align*} $$
Corollary 5.10. In the standard setting of convergence
$\Gamma _\infty $
is discrete and torsion-free.
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.
The next step is to show the stability of the boundary under convergence.
Proposition 5.11. Let
$(X_n,x_n)$
be a sequence of proper,
$\delta $
-hyperbolic metric spaces and let
$D_{x_n,a}$
be a standard visual metric of centre
$x_n$
and parameter a on
$\partial X_n$
. Let
$\omega $
be a non-principal ultrafilter and let
$(X_\omega , x_\omega )$
be the ultralimit of the sequence
$(X_n,x_n)$
. Then there exists a natural map
$\Psi \colon \omega $
-
$\lim (\partial X_n, D_{x_n,a}) \to \partial X_\omega $
which is a homeomorphism onto the image.
Remark 5.12. The following observations are in order.
-
(1) When the spaces
$\partial X_n$
are compact with diameter at most
$1$
, the ultralimit
$\omega $
-
$\lim \partial X_n$
does not depend on the basepoints. -
(2) In general the map
$\Psi $
is not surjective. Let
$X_n$
be the closed ball
$\overline {B}(o,n)$
inside the hyperbolic plane
$\mathbb {H}^2$
, where o is a fixed basepoint. Each
$X_n$
is proper and
$\delta $
-hyperbolic for the same
$\delta $
, but
$\partial X_n = \emptyset $
. Therefore
$\omega $
-
$\lim \partial X_n = \emptyset $
. On the other hand,
$X_\omega = \mathbb {H}^2$
and
$\partial X_\omega \neq \emptyset $
. -
(3) It is possible to prove (but we will not do so because it is not necessary for our purposes) that if for each point
$y_n$
of
$X_n$
there is a geodesic ray
$[x_n,z_n]$
passing at distance no greater than
$\delta $
from
$y_n$
then the map
$\Psi $
is surjective. Moreover, when
$\Psi $
is surjective, the metric induced on
$\partial X_\omega $
by
$\Psi $
is a visual metric of centre
$x_\omega $
and parameter a.
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:
-
(a)
$g_n^k x_n$
converges to
$z_n$
when k tends to
$+\infty $
; -
(b)
$g_n^0 = \text {id}$
; -
(c)
$d(g_n^k x_n, g_n^{k+1}x_n)\leq 28\delta + 2D$
; -
(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.
It is interesting to compare our convergence with the Gromov–Hausdorff convergence of the quotient spaces.
Theorem 5.13. Let
$(X_n,x_n,\Gamma _n), (X_\infty , x_\infty , \Gamma _\infty ) \in \mathcal {M}(\delta ,D)$
.
-
(i) If
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X_\infty , x_\infty ,\Gamma _\infty )$
then
$(\Gamma _n\backslash X_n, \bar {x}_n) \underset {\mathrm {pGH}}{\longrightarrow } (\Gamma _\infty \backslash X_\infty , \bar {x}_\infty )$
and
$\sup _{n\in \mathbb {N}}h_{\Gamma _n} < +\infty $
. -
(ii) If
$(\Gamma _n\backslash X_n, \bar {x}_n) \underset {\mathrm {pGH}}{\longrightarrow } (Y,y)$
and
$\sup _{n\in \mathbb {N}}h_{\Gamma _n} < +\infty $
then there exists a subsequence
$\lbrace n_k \rbrace $
such that
$(X_{n_k},x_{n_k},\Gamma _{n_k}) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X_\infty , x_\infty ,\Gamma _\infty )$
and
$(Y,y)$
is isometric to
$(\Gamma _\infty \backslash X_\infty , \bar {x}_\infty )$
.
Proof. If
$(X_n,x_n,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X_\infty , x_\infty ,\Gamma _\infty )$
then
$\sup _{n\in \mathbb {N}}h_{\Gamma _n} < +\infty $
by Corollary 5.9. The second part of the first statement is true once we show that the ultralimit
$(\bar X_\omega , \bar x_\omega )$
of the sequence
$(\Gamma _n\backslash X_n =:\bar X_n, \bar {x}_n)$
is isometric to
$(\Gamma _\infty \backslash X_\infty , \bar {x}_\infty )$
for every non-principal ultrafilter, by Corollary 3.14. We fix a non-principal ultrafilter
$\omega $
. By Proposition 3.13 the triple
$(X_\omega , x_\omega , \Gamma _\omega )$
is equivariantly isometric to
$(X_\infty , x_\infty , \Gamma _\infty )$
.
The projections
$p_n\colon X_n \to \bar X_n$
form an admissible sequence of
$1$
-Lipschitz maps and then, by [Reference Cavallucci and SambusettiCS21, Proposition A.5], they yield a limit map
$p_\omega \colon X_\omega \to \bar X_\omega $
defined as
$p_\omega ( y_\omega ) = \omega $
-
$\lim p_n( y_n)$
, for
$\omega $
-
$\lim y_n = y_\omega $
. The map
$p_\omega $
is clearly surjective. It is also
$\Gamma _\omega $
-equivariant: indeed,
$$ \begin{align*}p_\omega (\gamma_\omega y_\omega) = \omega\text{-}\!\lim p_n(\gamma_n y_n) = \omega\text{-}\!\lim p_n(y_n) = p_\omega(y_\omega)\end{align*} $$
for every
$g_\omega = \omega $
-
$\lim g_n \in \Gamma _\omega $
and
$y_\omega = \omega $
-
$\lim y_n \in X_\omega $
. Therefore we have a well-defined, surjective quotient map
$\bar {p}_\omega \colon \Gamma _\omega \backslash X_\omega \to \bar X_\omega $
. The next step is to show it is a local isometry. We fix an arbitrary point
$y_\omega = \omega $
-
$\lim y_n \in X_\omega $
and we consider its class
$[y_\omega ] \in \Gamma _\omega \backslash X_\omega $
. By Proposition 5.3 there exists
$s>0$
such that sys
$(\Gamma _n, X_n) \geq s$
for every n. So, as in the proof of Corollary 5.10, the systole of
$\Gamma _\omega $
is at least s. Therefore the quotient map
$X_\omega \to \Gamma _\omega \backslash X_\omega $
is an isometry between
$\overline {B}(y_\omega , {s}/{2})$
and
$\overline {B}([y_\omega ],{s}/{2})$
. Moreover,
$\overline {B}(p_n(y_n), {s}/{2})$
is isometric to
$\overline {B}(y_n,{s}/{2})$
for every n. By [Reference Cavallucci and SambusettiCS21, Lemma A.8] we know that
$\omega $
-
$\lim \overline {B}(p_n(y_n), {s}/{2})$
is isometric to
$\overline {B}(p_\omega (y_\omega ), {s}/{2})=\overline {B}(\bar p_\omega ([y_\omega ]), {s}/{2})$
and that
$\omega $
-
$\lim \overline {B}(y_n, {s}/{2})$
is isometric to
$\overline {B}(y_\omega , {s}/{2})$
. Therefore
$\overline {B}(\bar p_\omega ([y_\omega ]), {s}/{2})$
is isometric to
$\overline {B}([y_\omega ],{s}/{2})$
, that is,
$\bar p_\omega $
is a local isometry.
Now we prove that
$\bar {p}_\omega $
is injective. Let
$[z_\omega ], [y_\omega ] \in \Gamma _\omega \backslash X_\omega $
. Clearly
$\bar {p}_\omega ([z_\omega ]) = \bar {p}_\omega ([y_\omega ])$
if and only if
$p_\omega (z_\omega )=p_\omega (y_\omega )$
. This means
$\omega $
-
$\lim d(p_n (z_n), p_n(y_n)) = 0$
and, as the systole of
$\Gamma _n$
is greater than or equal to
$s>0$
, we have
$\omega $
-
$\lim d(z_n, g_n y_n) = 0$
for some
$g_n\in \Gamma _n$
,
$\omega $
-a.s. The sequence
$(g_n)$
is admissible, hence it defines an element
$g_\omega = \omega $
-
$\lim g_n \in \Gamma _\omega $
satisfying
$d(z_\omega ,g_\omega y_\omega )=0$
. This implies
$[z_\omega ] = [y_\omega ]$
.
The map
$\bar p_\omega \colon \Gamma _\omega \backslash X_\omega \to \bar X_\omega $
is a bijective local isometry between two length spaces. If its inverse is continuous then it is an isometry. We take points
$\bar y_\omega ^k = \omega $
-
$\lim \bar y_n^k \in \bar X_\omega $
converging to
$\bar y_\omega ^\infty = \omega $
-
$\lim \bar y_n^\infty \in \bar X_\omega $
as
$k \to +\infty $
. We have
$\bar y_n^k = p_n(y_n^k), \bar y_n^\infty = p_n(y_n^\infty )$
for some
$y_n^k,y_n^\infty \in X_n$
. We can suppose that
$y_n^k,y_n^\infty $
belong to a fixed ball around
$x_n$
. We consider the points
$y_\omega ^k = \omega $
-
$\lim y_n^k$
and
$y_\omega ^\infty = \omega $
-
$\lim y_n^\infty $
of
$X_\omega $
and their images
$[y_\omega ^k], [y_\omega ^\infty ] \in \Gamma _\omega \backslash X_\omega $
. It is straightforward to show that
$\bar p_\omega ([y_\omega ^k]) = \bar y_\omega ^k$
and
$\bar p_\omega ([y_\omega ^\infty ]) = \bar y_\omega ^\infty $
. Now it is not difficult to check that the sequence
$[y_\omega ^k]$
converges to
$[y_\omega ^\infty ]$
when
$k \to +\infty $
, proving that the inverse of
$\bar {p}_\omega $
is continuous.
Therefore
$(\bar X_\omega , \bar {x}_\omega )$
is isometric to
$(\Gamma _\omega \backslash X_\omega , p_\omega {x_\omega })$
which is clearly isometric to
$(\Gamma _\infty \backslash X_\infty , \bar {x}_\infty )$
. The proof of (i) is then finished since this is true for every non-principal ultrafilter
$\omega $
.
Suppose now that
$(\Gamma _n\backslash X_n, \bar {x}_n) \underset {\mathrm {pGH}}{\longrightarrow } (Y,y)$
and
$\sup _{n\in \mathbb {N}}h_{\Gamma _n} < +\infty $
. Again by Proposition 5.3 there exists
$s>0$
such that sys
$(\Gamma _n, X_n) \geq s$
for every n. We fix a non-principal ultrafilter
$\omega $
and we consider the ultralimit triple
$(X_\omega ,x_\omega ,\Gamma _\omega )$
. As usual, we get sys
$(\Gamma _\omega ,X_\omega ) \geq s$
. We can apply the same argument above to show that
$\Gamma _\omega \backslash X_\omega $
is isometric to Y. Moreover, by the condition on the systole of
$\Gamma _\omega $
we know that
$X_\omega $
is locally isometric to
$\Gamma _\omega \backslash X_\omega $
. Since Y is compact we conclude that
$X_\omega $
is a geodesic, complete, locally compact metric space. Therefore it is proper by the Hopf–Rinow theorem; see [Reference Bridson and HaefligerBH13, Corollary I.3.8]. Then there exists a subsequence
$\lbrace n_k \rbrace $
such that
$(X_{n_k},x_{n_k},\Gamma _{n_k}) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X_\omega , x_\omega ,\Gamma _\omega )$
by Proposition 3.13.
6 Continuity of the critical exponent
Let
$\Gamma $
be a discrete, quasiconvex-cocompact group of isometries of a proper,
$\delta $
-hyperbolic metric space X. Then it is proved in [Reference CoornaertCoo93] that the Patterson–Sullivan measure on
$\Lambda (\Gamma )$
is
$(A,h_\Gamma )$
-Ahlfors regular for some
$A> 0$
. Our goal is to quantify the constant A.
Theorem 6.1. Let
$\delta ,D,H\geq 0$
. There exists
$A=A(\delta ,D,H) \geq 1$
such that for all
$(X,x,\Gamma )\in \mathcal {M}(\delta ,D)$
with
$h_\Gamma \leq H$
the subset
$\Lambda (\Gamma )$
is visually
$(A,h_\Gamma )$
-Ahlfors regular with respect to every Patterson–Sullivan measure.
Proof. We divide the proof into steps.
Step 1:
$\text { for all } z\in \partial X$
and
$\text { for all }\rho>0$
we have
$\mu _{\mathrm {PS}}(B(z,\rho ))\leq e^{h_\Gamma (55\delta + 3D)}\rho ^{h_\Gamma }$
.
We suppose first
$z\in \Lambda (\Gamma )$
and we take the set
$$ \begin{align*}\tilde{B}(z,\rho)= \bigg\lbrace y \in X \cup \partial X \text{ s.t. } (y,z)_{x}> \log\frac{1}{\rho} \bigg\rbrace.\end{align*} $$
It is open (cf. [Reference Das, Simmons and UrbańskiDSU17, Observation 4.5.2]) and
$\tilde {B}(z,\rho )\cap \partial X = B(z,\rho )$
, so
$\mu _{\text {PS}}(\tilde {B}(z,\rho )) = \mu _{\text {PS}}(B(z,\rho ))$
since
$\mu _{\text {PS}}$
is supported on
$\Lambda (\Gamma )\subseteq \partial X$
. Let
$T=\log ({1}/{\rho })$
and
$\xi _{x,z}$
be a geodesic ray
$[x,z]$
. For every
$y\in \Gamma x \cap \tilde {B}(z,\rho )$
we have
$$ \begin{align} d(x,y)\geq T - \delta \quad\text{and}\quad d(x,y) \geq d(x,\xi_{x,z}(T)) + d(\xi_{x,z}(T), y) - 12\delta. \end{align} $$
The first inequality is given by Remark 4.3. Let
$\gamma $
be any geodesic segment
$[x,y]$
. Again by Remark 4.3 we have
$d(\xi _{x,z}(T), \gamma (T)) \leq 6\delta $
, therefore
$$ \begin{align*}d(x,y) = d(x,\gamma(T)) + d(\gamma(T), y) \geq d(x,\xi_{x,z}(T)) + d(\xi_{x,z}(T), y) -12\delta.\end{align*} $$
Moreover, we have
$d(\xi _{x,z}(T), \textrm{QC-Hull}(\Lambda (\Gamma ))) \leq 14\delta $
by Lemma 4.6, since
$x\in \textrm{QC-Hull}(\Lambda (\Gamma ))$
. By the cocompactness of the action on
$\textrm{QC-Hull}(\Lambda (\Gamma ))$
we can find a point
$x_1 \in \Gamma x$
such that
$d(\xi _{x,z}(T), x_1)\leq 14\delta +D$
. This implies
$$ \begin{align*}d(x,y) \geq d(x,x_1) + d(x_1, y) - 40\delta - 2D\end{align*} $$
for every
$y\in \Gamma x \cap \tilde {B}(z,\rho )$
. Therefore
$$ \begin{align*} \begin{aligned} \sum_{y \in \Gamma x \cap \tilde{B}(z,\rho)} e^{-s d(x,y)} &\leq \sum_{y \in \Gamma x \cap \tilde{B}(z,\rho)} e^{-s (d(x, x_1) + d(x_1,y) - 40 \delta - 2D)} \\ &= e^{s(40\delta + 2D)}e^{-sd(x,x_1)}\cdot \sum_{y \in \Gamma x \cap \tilde{B}(z,\rho)}e^{-sd(x_1,y)} \\ &\leq e^{s(54\delta + 3D)}e^{-sd(x,\xi_{xz}(T))}\cdot \sum_{g \in \Gamma}e^{-sd(x_1,g x_1)} \\ &= e^{s(54\delta + 3D)}\cdot\rho^s\cdot \sum_{g \in \Gamma}e^{-sd(x,g x)}. \end{aligned} \end{align*} $$
In other words, we have
$\mu _s(\tilde {B}(z,\rho )) \leq e^{s(54\delta + 3D)}\rho ^s$
, and by
$\ast $
-weak convergence we conclude that
$$ \begin{align*} \begin{aligned} \mu_{\text{PS}}(B(z,\rho)) = \mu_{\text{PS}}(\tilde{B}(z,\rho)) \leq \liminf_{i\to +\infty} \mu_{s_i}(\tilde{B}(z,\rho)) \leq e^{h_\Gamma(54\delta + 3D)}\rho^{h_\Gamma}. \end{aligned} \end{align*} $$
If
$z\in \partial X$
we observe that if
$B(z,\rho )\cap \Lambda (\Gamma )=\emptyset $
then the thesis is obviously true since
$\mu _{\text {PS}}$
is supported on
$\Lambda (\Gamma )$
. Otherwise there exists
$w\in \Lambda (\Gamma )$
such that
$(z,w)_{x}> \log ({1}/{\rho })$
. It is straightforward to check that
$B(w,\rho ) \subseteq B(z,\rho e^\delta )$
by (4.3). Then
$\mu _{\text {PS}}(B(z,\rho ))\leq e^{h_\Gamma (55\delta + 3D)}\rho ^{h_\Gamma }$
.
Step 2: for every
$R\geq R_0 :=({\log 2}/{h_\Gamma }) + 55\delta + 3D + 5\delta $
and for every
$g \in \Gamma $
we have
$\mu _{\mathrm {PS}}(\mathrm {Shad}_{x}(g x, R)) \geq ({1}{/2Q})e^{-h_\Gamma d(x,g x)},$
where Q is the constant of Proposition 4.13 that depends only on
$\delta $
and H.
From the first step we know that for every
$\rho \leq \rho _0 := 2^{-({1}/{h_\Gamma })}e^{-(55\delta + 3D)}$
and for every
$z\in \partial X$
the inequality
$\mu _{\text {PS}}(B(z,\rho ))\leq \tfrac 12$
holds. A direct computation shows that
$R_0 = \log ({1}/{\rho _0}) + 5\delta $
. We claim that for every
$R\geq R_0$
and every
$g \in \Gamma $
the set
$\partial X \setminus g(\text {Shad}_{x}(g^{-1}x,R))$
is contained in a generalized visual ball of radius at most
$\rho _0$
. Indeed, if
$z,w \in \partial X \setminus g(\text {Shad}_{x}(g^{-1}x,R))$
then there are geodesic rays
$\xi =[g x, z], \xi '=[g x, w]$
that do not intersect the ball
$B(x,R)$
. Therefore we get
$(\xi (T),g x)_{x} \geq d(x, [g x, \xi (T)]) - 4\delta \geq R - 4\delta $
by Lemma 4.1, so
$(z,g x)_{x} \geq \liminf _{T\to +\infty } (\xi (T),g x)_{x} \geq R - 4\delta $
. The same holds for w. Thus by (4.3) we get
$(z,w)_{x} \geq R - 5\delta ,$
proving the claim. By Proposition 4.13 we get
$$ \begin{align*}\frac{\mu_{\text{PS}}(\text{Shad}_{x}(g x, R))}{\mu_{\text{PS}}(g^{-1}(\text{Shad}_{x}(g x, R)))} \geq \frac{1}{Q}e^{-h_\Gamma (B_z(x,x) - B_z(x,g^{-1} x))}.\end{align*} $$
Since
$R\geq R_0$
the measure of
$g^{-1}(\text {Shad}_{x}(g x, R))$
is at least
$\tfrac 12$
. Moreover, the Busemann function is
$1$
-Lipschitz, so
$$ \begin{align*}\mu_{\text{PS}}(\text{Shad}_{x}(g x, R)) \geq \frac{1}{2Q}e^{-h_\Gamma d(x,g^{-1}x)} = \frac{1}{2Q}e^{-h_\Gamma d(x,gx)}.\end{align*} $$
Step 3.
$\mu _{\mathrm {PS}}(B(z,\rho ))\geq ({1}/{2Q})e^{-h_\Gamma (R_0+28\delta +2D)}\rho ^{h_\Gamma }$
for every
$z\in \Lambda (\Gamma )$
and every
$\rho>0$
.
For every
$\rho> 0$
we set
$T=\log ({1}/{\rho })$
. If
$z\in \partial X$
and
$R\geq 0$
then by Lemma 4.8 we get Shad
$_{x}(\xi _{x,z}(T+R), R) \subseteq B(z,e^{-T})$
. We take
$R=R_0 + 14\delta + D$
, where
$R_0$
is the constant of the second step, and we conclude that Shad
$_{x}(\xi _{x,z}(T+R), R)$
is contained in
$B(z,\rho )$
. Again applying Lemma 4.6 and the cocompactness of the action, we can find
$g \in \Gamma $
such that
$d(gx, \xi _{x,z}(T+R))\leq 14\delta + D$
, implying Shad
$_{x}(gx,R_0)\subseteq \text {Shad}_{x}(\xi _{x,z}(T+R), R) \subseteq B(z,\rho )$
. From the second step we obtain
$\mu _{\text {PS}}(B(z,\rho )) \geq ({1}/{2Q})e^{-h_\Gamma d(x,g x)}.$
Furthermore,
$d(x, g x) \leq T + R_0 + 28\delta + 2D,$
so finally
$$ \begin{align*}\mu_{\text{PS}}(B(z,\rho))\geq \frac{1}{2Q}e^{-h_\Gamma(R_0+28\delta +2D)}\rho^{h_\Gamma}.\end{align*} $$
The explicit description of the constants shows that they depend only on
$\delta ,H,D$
and on a lower bound on
$h_\Gamma $
, which is given in terms of
$\delta $
and D by Proposition 5.2.
As a consequence, applying Corollary 5.9, we have the following result.
Corollary 6.2. In the standard setting of convergence there exists some
$A>0$
such that every visual boundary
$\partial X_n$
is visually
$(A,h_{\Gamma _n})$
-Ahlfors-regular with respect to any Patterson–Sullivan measure.
With this result it is possible to show the continuity of the critical exponent under the standard setting of convergence. However, we prefer to use the equidistribution of the orbits, following again the ideas of [Reference CoornaertCoo93].
Proof of Theorem B
By Corollary 5.9 we have
$\sup _{n\in \mathbb {N}}h_{\Gamma _n} =: H < + \infty $
. By Proposition 5.3 there exists
$s> 0$
such that
$\text {sys}(\Gamma _n,X_n) \geq s$
for every n. Let
$R_0 = R_0(\delta , D, H)$
be the number from Step 2 of Theorem 6.1 and Q be the constant from Proposition 4.13. By Lemma 5.8 we have
$$ \begin{align*}\sup_{n\in\mathbb{N}}\text{Pack}_{Y_n}\bigg(4R_0+1,\frac{s}{2}\bigg) =: N <+\infty.\end{align*} $$
We fix
$n \in \mathbb {N}$
. It is easy to check that if
$[x_n,z_n]\cap B(y_n,R_0) \neq \emptyset $
and
$[x_n,z_n]\cap B(y^{\prime }_n,R_0) \neq \emptyset $
, where
$z_n\in \partial X$
and
$y_n,y^{\prime }_n$
are points of
$X_n$
with
$\vert d(x_n,y_n) - d(x_n,y^{\prime }_n)\vert \leq 1$
, then
$d(y_n,y^{\prime }_n)\leq 4R_0 + 1$
. Thus for every
$j\in \mathbb {N}$
we have
$\#\lbrace y_n\in \Gamma _n x_n \text { s.t. } y_n\in A(x_n,j,j+1) \text { and } z_n\in \mathrm {Shad}_{x_n}(y_n,R_0) \rbrace \leq N.$
Step 1. For all
$k\in \mathbb {N}$
we have
$\#\Gamma _n x_n \cap \overline {B}(x_n,k) \leq 4QNe^{h_{\Gamma _n} k}.$
Let
$A_{n,j} = \Gamma _n x_n \cap A(x_n,j,j+1)$
. By the observation made before, we conclude that among the set of shadows
$\lbrace \text {Shad}_{x_n}(y_n,R_0) \rbrace _{y_n\in A_j}$
there are at least
${\# A_j}/{N}$
disjoint sets. Thus
$$ \begin{align*}1 \geq \mu_{\text{PS}}\bigg( \bigcup_{y_n\in A_{n,j}}\text{Shad}_{x_n}(y_n,R_0) \bigg) \geq \frac{\# A_{n,j}}{N}\cdot \frac{1}{2Q}e^{-h_{\Gamma_n}(j+1)},\end{align*} $$
where we used Step 2 of Theorem 6.1. This implies
$\#A_{n,j} \leq 2QNe^{h_{\Gamma _n}(j+1)}$
for every
$j\in \mathbb {N}$
. Finally, we have
$$ \begin{align*}\#\Gamma_n x_n \cap \overline{B}(x_n,k) \leq \sum_{j=0}^{k-1} \#A_{n,j} \leq 4QNe^{h_{\Gamma_n} k}.\end{align*} $$
Step 2. For all
$T\geq 0$
we have
$\#\Gamma _n x_n \cap \overline {B}(x_n,T) \geq e^{-h_{\Gamma _n}(84\delta + 5D + 1)}e^{h_{\Gamma _n} T}.$
We fix
$z_1^n,\ldots ,z_{K_n}^n \in \Lambda (\Gamma _n)$
, realizing Pack
$^*(\Lambda (\Gamma _n), e^{-T + 28\delta + 2D + 1})$
: in particular,
$(z_i^n,z_j^n)_{x_n} \leq T - 28\delta - 2D - 1$
for all
$1\leq i \neq j \leq K_n$
. By Lemma 4.2 we deduce that
$d(\xi _{x_n,z_i^n}(T-14\delta -D), \xi _{x_n,z_j^n}(T-14\delta -D)) \geq 28\delta + 2D + 1> 28\delta + 2D$
. Moreover, for every
$1\leq i \leq K_n$
we can find a point
$y_i^n\in \Gamma x$
such that
$d(\xi _{x_n,z_i^n}(T-14\delta -D), y_i^n) \leq 14\delta + D$
by Lemma 4.6. Therefore we have
$d(x_n,y_i^n)\leq T$
and
$d(y_i^n,y_j^n)> 0$
for every
$1\leq i \neq j \leq K_n$
. So
$$ \begin{align*} \begin{aligned} \#\Gamma_n x_n \cap \overline{B}(x_n,T) &\geq \text{Pack}^*(\Lambda(\Gamma_n), e^{-T+28\delta + 2D + 1}) \\ &\geq \text{Cov}(\Lambda(\Gamma_n), e^{-T+29\delta + 2D + 1}) \\ &\geq e^{-h_{\Gamma_n}(84\delta + 5D + 1)}e^{h_{\Gamma_n} T}. \end{aligned} \end{align*} $$
The first inequality follows from the discussion above, while the second is Lemma 4.10. The last inequality follows by Step 1 of Theorem 6.1. Indeed, we get
$$ \begin{align*} \begin{aligned} \text{Cov}(\Lambda(\Gamma_n), e^{-T + 29\delta + 2D + 1}) &\geq e^{-h_{\Gamma_n}(55\delta + 3D)}e^{-h_{\Gamma_n}(-T + 29\delta + 2D + 1)} \\ &=e^{-h_{\Gamma_n}(84\delta + 5D + 1)}e^{h_{\Gamma_n} T}. \end{aligned} \end{align*} $$
The thesis follows by the bounded quantification of all the constants involved in terms of
$\delta , D, H, N$
and the lower bound on the critical exponent given in terms of
$\delta $
and D by Proposition 5.2.
We can conclude now the proof of Theorem A.
Proof of Theorem A(ii)
Let
$\omega $
be a non-principal ultrafilter. By Proposition 3.13 the triple
$(X_\omega ,x_\omega , \Gamma _\omega )$
is equivariantly isometric to
$(X_\infty , x_\infty , \Gamma _\infty )$
. By Theorem A(i) the triple
$(X_\omega ,x_\omega , \Gamma _\omega )$
belongs to
$\mathcal {M}(\delta ,D)$
. By Proposition 5.7 the critical exponent of
$\Gamma _\omega $
is finite, so for every
$\varepsilon> 0$
there exists
$T_\varepsilon \geq 0$
such that if
$T\geq T_\varepsilon $
then
$$ \begin{align} e^{T(h_{\Gamma_\omega} -\varepsilon)} \leq \#\Gamma_\omega x_\omega\cap\overline{B}(x_\omega,T) \leq e^{T(h_{\Gamma_\omega} +\varepsilon)}, \end{align} $$
by Lemma 4.12. We fix K as in Theorem B and we set
$T:= \max \lbrace T_\varepsilon , {\log {K\cdot e}}/{\varepsilon }\rbrace $
. It is not difficult to show that
$$ \begin{align*}\#\Gamma_nx_n\cap\overline{B}(x_n,T-1) \leq \#\Gamma_\omega x_\omega\cap\overline{B}(x_\omega,T) \leq \#\Gamma_nx_n\cap\overline{B}(x_n,T+1)\end{align*} $$
$\omega $
-a.s., so
$$ \begin{align} \frac{1}{K}\cdot e^{-1}\cdot e^{T\cdot h_{\Gamma_n}} \leq \#\Gamma_\omega x_\omega\cap\overline{B}(x_\omega,T) \leq K \cdot e \cdot e^{T\cdot h_{\Gamma_n}} \end{align} $$
$\omega $
-a.s. Putting together (6.2) and (6.3) and using the definition of T, we get
$$ \begin{align*}h_{\Gamma_n} - 2\varepsilon \leq h_{\Gamma_\omega} \leq h_{\Gamma_n} + 2\varepsilon,\end{align*} $$
$\omega $
-a.s. This means
$\omega $
-
$\lim h_{\Gamma _n} = h_{\Gamma _\omega }$
, by definition. This is true for every non-principal ultrafilter, hence the continuity under equivariant pointed Gromov–Hausdorff convergence follows by the next result.
Lemma 6.3. Let
$a_n$
be a bounded sequence of real numbers.
-
(i) If
$a_{n_j}$
is a subsequence converging to
$\tilde {a}$
then there exists a non-principal ultrafilter
$\omega $
such that
$\omega $
-
$\lim a_n = \tilde {a}$
. -
(ii) If there exists
$a\in \mathbb {R}$
such that
$\omega $
-
$\lim a_n = a$
for every non-principal ultrafilter
$\omega $
, then
$\text { there exists } \lim _{n\to +\infty } a_n = a$
.
Proof. Let us start with (i). The set
$\{n_j\}_j$
is infinite, so there exists a non-principal ultrafilter
$\omega $
containing
$\{n_j\}_j$
(cf. [Reference JansenJan17, Lemma 3.2]). Moreover, for every
$\varepsilon> 0$
there exists
$j_\varepsilon $
such that for all
$j\geq j_\varepsilon $
the inequality
$\vert a_{n_j} - \tilde {a} \vert < \varepsilon $
holds. The set of indices where the inequality is true belongs to
$\omega $
since the complementary is finite. This implies exactly that
$\tilde {a} = \omega $
-
$\lim a_n$
.
The proof of (ii) is now a direct consequence. We take subsequences
$\lbrace n_j\rbrace _{j\in J}$
,
$\lbrace n_k\rbrace _{k\in K}$
converging respectively to the limit inferior and limit superior of the sequence. By (i) there are two non-principal ultrafilters
$\omega _J$
,
$\omega _K$
such that
$\omega _J$
-
$\lim a_{n_j} = \liminf _{n\to +\infty }a_n$
and
$\omega _K$
-
$\lim a_{n_k} = \limsup _{n\to +\infty }a_n$
. By assumption these two ultralimits coincide, so
$\liminf _{n\to +\infty }a_n = \limsup _{n\to +\infty }a_n$
.
7 Algebraic and equivariant Gromov–Hausdorff convergence
Let X be a proper metric space and G be a topological group. We denote by Act
$(G,X)$
the set of homomorphisms
$\varphi \colon G \to \text {Isom}(X)$
.
Definition 7.1. Let
$\varphi _n,\varphi _\infty \in \mathrm {Act}(G,X)$
. We say
$\varphi _n$
converges in the algebraic sense to
$\varphi _\infty $
if
$\varphi _n$
converges to
$\varphi _\infty $
in the compact-open topology. In this case we write
$\varphi _n \underset {\mathrm {alg}}{\longrightarrow } \varphi _\infty $
. The compact-open topology is Hausdorff since Isom
$(X)$
is, so the algebraic limit is unique, if it exists.
If G has the discrete topology then the algebraic convergence is equivalent to the isometries
$\varphi _n(g)$
converging to
$\varphi _\infty (g)$
uniformly on compact subsets of X for every
$g\in G$
.
If G has the discrete topology and is finitely generated by
$\lbrace g_1,\ldots ,g_\ell \rbrace $
then the algebraic convergence is equivalent to the isometries
$\varphi _n(g_i)$
converging to
$\varphi _\infty (g_i)$
uniformly on compact subsets of X for every
$i=1,\ldots ,\ell $
.
The algebraic limit is always contained in the ultralimit group in the following sense.
Proposition 7.2. Let
$\varphi _n,\varphi _\infty \in \mathrm {Act}(G,X)$
and suppose
$\varphi _n \underset {\mathrm {alg}}{\longrightarrow }{\varphi _\infty }$
. Let
$\omega $
be a non-principal ultrafilter. Then
$\varphi _\infty (G) \subseteq (\varphi _n(G))_\omega $
, where we use Lemma 3.9 to identify the ultralimit group of the sequence
$\varphi _n(G)$
with a group of isometries of X.
Proof. For every
$g\in G$
the sequence
$\varphi _n(g)$
converges uniformly on compact subsets of X to
$\varphi _\infty (g)$
by assumption. It is easy to see that the ultralimit element
$\omega $
-
$\lim \varphi _n(g)$
coincides with
$\varphi _\infty (g)$
, so
$ \varphi _\infty (G) \subseteq (\varphi _n(G))_\omega $
.
In general the inclusion is strict.
Example 7.3. Let
$X=\mathbb {R}, G = \mathbb {Z}$
and
$\varphi _n \colon \mathbb {Z} \to \mathrm {Isom}(\mathbb {R})$
defined by sending
$1$
to the translation of length
${1}/{n}$
. Clearly the sequence
$\varphi _n$
converges algebraically to the trivial homomorphism. On the other hand,
$(\varphi _n(\mathbb {Z}))_\omega $
is the group
$\Gamma $
of all translations of
$\mathbb {R}$
for every non-principal ultrafilter
$\omega $
.
However, the two limits coincide when restricted to the class
$\mathcal {M}(\delta ,D)$
.
Theorem 7.4. Let
$(X,x, \Gamma _n)\in \mathcal {M}(\delta ,D)$
.
-
(i) If
$(X,x, \Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow }(X,x, \Gamma _\infty )$
then there exists a group G such that, for every sufficiently large n, we have
$\Gamma _{n} = \varphi _{n}(G)$
,
$\Gamma _\infty = \varphi _\infty (G)$
with
$\varphi _{n},\varphi _\infty $
isomorphisms and
$\varphi _{n} \underset {\mathrm {alg}}{\longrightarrow } \varphi _\infty $
. -
(ii) Conversely, if there exist a group G and isomorphisms
$\varphi _n\colon G \to \Gamma _n$
and if
$\varphi _n \underset {\mathrm {alg}}{\longrightarrow } \varphi _\infty $
then
$(X,x,\Gamma _n) \underset {\mathrm {eq-pGH}}{\longrightarrow } (X,x,\varphi _\infty (G))$
.
Before the proof we need two results. Given a group
$\Gamma $
and a finite set of generators
$\Sigma $
of
$\Gamma $
, the word-metric
$d_\Sigma $
is classically defined on
$\Gamma $
as
$$ \begin{align*}d_\Sigma(g,h) := \inf\lbrace \ell \in \mathbb{N} \text{ s.t. } g = h \cdot \sigma_1\cdots\sigma_\ell, \text{ where } \sigma_i\in \Sigma\rbrace.\end{align*} $$
By definition
$d_\Sigma $
takes values in the set of natural numbers and
$d_\Sigma (g,h)=d_\Sigma (h^{-1}g, \text {id})$
. The couple
$(\Gamma ,\Sigma )$
is called a marked group.
Lemma 7.5. [Reference Besson, Courtois, Gallot and SambusettiBCGS21, Lemma 4.6]
Let
$(X,x,\Gamma )\in \mathcal {M}(\delta ,D)$
and let
$R> 2D + 72\delta $
. Set
$\Sigma :=\Sigma _R(\Gamma ,x)$
and denote by
$d_\Sigma $
the associated word-metric on
$\Gamma $
(observe that
$\Sigma $
is a generating set of
$\Gamma $
by Lemma 5.1). Then
$$ \begin{align*}(R - 2D - 72\delta) \cdot d_\Sigma(g,h) \leq d(gx,hx) \leq R \cdot d_\Sigma(g,h)\end{align*} $$
for all
$g,h\in \Gamma $
.
Proof. It is enough to check the inequalities for
$g\in \Gamma $
and
$h=\text {id}$
. We write
$g=\sigma _1\cdots \sigma _\ell $
, with
$\sigma _i \in \Sigma $
,
$\ell = d_\Sigma (g,\text {id})$
. The right inequality follows by the triangle inequality on X, indeed
$d(gx,x)\leq \ell \cdot R$
.
We now take consecutive points
$x_i$
,
$i=0,\ldots ,\ell $
, along a geodesic segment
$[x,gx]$
with
$x_0=x$
,
$x_\ell = gx$
,
$d(x_{i-1},x_{i}) = R-2D-72\delta $
for
$i=1,\ldots ,\ell -1$
and
$d(x_{\ell -1}, gx) \leq R-2D-72\delta $
. This implies
$\ell \leq {d(x,gx)}/({R-2D-72\delta })$
. By Lemma 4.5 and by cocompactness we can find an element
$g_i \in \Gamma $
such that
$d(x_i,g_ix)\leq 36\delta + D$
for every
$i=0,\ldots ,\ell $
. We choose
$g_0=\text {id}$
and
$g_\ell = g$
. Clearly
$d(g_ix, g_{i-1}x) \leq R$
for every
$i=1,\ldots ,\ell $
. This shows that
$\sigma _i = g_{i-1}^{-1}g_i \in \Sigma $
. Moreover,
$g = \sigma _1\cdots \sigma _\ell $
, that is,
$d_{\Sigma }(g,\text {id}) \leq \ell \leq {d(x,gx)}/({R-2D-72\delta })$
.
In the following proposition we make the metric in the pointed Gromov–Hausdorff convergence explicit for the sake of clarity.
Proposition 7.6. In the standard setting of convergence let R be a real number satisfying:
-
(i)
$2D+72\delta < R \leq 2D+72\delta +1$
; -
(ii) for every
$g\in \Gamma _\infty $
such that
$d(x_\infty ,gx_\infty )\leq R$
, we have
$d(x_\infty , gx_\infty ) < R$
.
Let
$\Sigma _n := \Sigma _R(\Gamma _n,x_n)$
and
$\Sigma _\infty = \Sigma _R(\Gamma _\infty , x_\infty )$
be generating sets of
$\Gamma _n$
and
$\Gamma _\infty $
, respectively (by Lemma 5.1). Equip
$\Gamma _n$
and
$\Gamma _\infty $
with the word-metrics
$d_{\Sigma _n}$
,
$d_{\Sigma _\infty }$
, respectively. Then
$(\Gamma _n, d_{\Sigma _n}, \mathrm {id}) \underset {\mathrm {pGH}}{\longrightarrow } (\Gamma _\infty , d_{\Sigma _\infty }, \mathrm {id}).$
Proof. We fix a non-principal ultrafilter
$\omega $
, we take the ultralimit triple
$(X_\omega ,x_\omega , \Gamma _\omega )$
and we set
$\Sigma _\omega := \Sigma _R(\Gamma _\omega ,x_\omega )$
. We have that
$(X_\omega , x_\omega , \Gamma _\omega )$
is equivariantly isometric to
$(X_\infty ,x_\infty ,\Gamma _\infty )$
by Proposition 3.13, so
$(\Gamma _\omega , d_{\Sigma _\omega }, \text {id})$
is isometric to
$(\Gamma _\infty , d_{\Sigma _\infty }, \text {id})$
, and they are proper. If we show that the ultralimit of the sequence of spaces
$(\Gamma _n, d_{\Sigma _n}, \text {id})$
is isometric to
$(\Gamma _\omega , d_{\Sigma _\omega }, \text {id})$
we conclude by Corollary 3.14 that
$(\Gamma _n, d_{\Sigma _n}, \mathrm {id}) \underset {\mathrm {pGH}}{\longrightarrow } (\Gamma _\infty , d_{\Sigma _\infty }, \mathrm {id})$
.
We denote by
$\omega $
-
$\lim (\Gamma _n, d_{\Sigma _n}, \text {id})$
the ultralimit space of this sequence. Observe that each element is represented by a sequence
$(g_n)$
with
$g_n\in \Gamma _n$
and
$d_{\Sigma _n}(\text {id},g_n) \leq M \omega $
-a.s., for some M. We define
$\Phi \colon \omega $
-
$\lim (\Gamma _n, d_{\Sigma _n}, \text {id}) \to (\Gamma _\omega , d_{\Sigma _\omega }, \text {id})$
by sending a point
$(g_n)$
to the isometry of
$\Gamma _\omega $
defined by
$\omega $
-
$\lim g_n$
. We have to show that
$\Phi $
is well defined. First of all the condition
$d_{\Sigma _n}(\text {id},g_n) \leq M$
implies
$d(x_n,g_nx_n)\leq R\cdot M \omega $
-a.s. by Lemma 7.5, so the sequence
$(g_n)$
is admissible and defines a limit isometry belonging to
$\Gamma _\omega $
. Suppose that
$(h_n)$
is another sequence of isometries of
$\Gamma _n$
such that
$\omega $
-
$\lim d_{\Sigma _n}(g_n,h_n) = 0$
. Then
$d_{\Sigma _n}(g_n,h_n) < 1 \omega $
-a.s, thus
$d_{\Sigma _n}(g_n,h_n) = 0 \omega $
-a.s., that is,
$g_n = h_n \omega $
-a.s. This shows that the map
$\Phi $
is well defined.
It remains to show that it is an isometry. Let
$(g_n), (h_n) \in \omega $
-
$\lim (\Gamma _n, d_{\Sigma _n}, \text {id})$
and set
$\ell = \omega $
-
$\lim d_{\Sigma _n}(g_n,h_n)$
. Since any word-metric takes values in
$\mathbb {N}$
we get
$d_{\Sigma _n}(g_n,h_n) = \ell \omega $
-a.s. For these indices we can write
$g_n = h_n \cdot a_n^1 \cdots a_n^\ell $
with
$a_n^1,\ldots ,a_n^\ell \in \Sigma _n$
. The sequence of isometries
$(a_n^i)$
are admissible by definition, so they define elements
$a_\omega ^i \in \Sigma _\omega $
. We have
$g_\omega = h_\omega \cdot a_\omega ^1 \cdots a_\omega ^\ell $
, showing that
$d_{\Sigma _\omega }(g_\omega ,h_\omega ) \leq \ell = \omega $
-
$\lim d_{\Sigma _n}(g_n,h_n)$
.
We now take isometries
$g_\omega =\omega $
-
$\lim g_n$
,
$h_\omega = \omega $
-
$\lim h_n$
of
$\Gamma _\omega $
. By definition
$d(x_n,g_nx_n) \leq M$
and
$d(x_n,h_nx_n) \leq M\ \omega $
-a.s., for some M. By Lemma 7.5 we get
$d_{\Sigma _n}(g_n,\text {id}) \leq M'$
and
$d_{\Sigma _n}(h_n,\text {id}) \leq M'\ \omega $
-a.s., for
$M' = {M}/({R-2D-72\delta })$
. Therefore the sequences
$(g_n),(h_n)$
defines point of
$\omega $
-
$\lim (\Gamma _n, d_{\Sigma _n}, \text {id})$
. Observe that this shows also that
$\Phi $
is surjective. We set
$d_{\Sigma _\omega }(g_\omega ,h_\omega ) = \ell $
. Then we can write
$g_\omega = h_\omega \cdot a_\omega ^1\cdots a_\omega ^\ell $
, for some
$a_\omega ^i = \omega $
-
$\lim a_n^i \in \Sigma _\omega $
. This means that
$d(x_\omega , a_\omega ^i x_\omega ) \leq R$
, so
$d(x_\omega , a_\omega ^i x_\omega ) < R$
by our assumptions on R. Therefore the following finite set of conditions holds
$\omega $
-a.s.:
$d(x_n, a_n^i x_n) \leq R$
for every
$i=1,\ldots ,\ell $
, that is,
$a_n^i \in \Sigma _n \omega $
-a.s. Now observe that if
$g_n \neq h_n\cdot a_n^1\cdots a_n^\ell =: b_n \omega $
-a.s. then
$d(g_nx_n,b_nx_n)\geq s> 0\ \omega $
-a.s. Indeed, by Corollary 5.9 and Proposition 5.3 it is enough to take s smaller than a uniform lower bound of the systole of all the groups
$\Gamma _n$
. Hence we get
$d(g_\omega x_\omega , b_\omega x_\omega )> 0$
, which is clearly false. This shows that
$\omega $
-
$\lim d_{\Sigma _n}(g_n,h_n)\leq \ell $
, that is,
$d_{\Sigma _\omega }(g_\omega , h_\omega ) \geq \omega $
-
$\lim d_{\Sigma _n}(g_n,h_n)$
. Therefore
$\Phi $
is an isometry.
Proof of Theorem 7.4
We first prove (i). We can always find R as in the assumptions of Proposition 7.6 since
$\Gamma _\infty $
is discrete by Corollary 5.10. So, with the same notation as in Proposition 7.6,
$(\Gamma _n, d_{\Sigma _n}, \mathrm {id}) \underset {\mathrm {pGH}}{\longrightarrow } (\Gamma _\infty , d_{\Sigma _\infty }, \mathrm {id}).$
Applying word for word the proof of Theorem 4.4 of [Reference Besson, Courtois, Gallot and SambusettiBCGS21], using Lemma 7.5 instead of Lemma 4.6 therein, we get only a finite number of isomorphic types of the marked groups
$(\Gamma _n, \Sigma _n)$
. This implies that for sufficiently large n all the marked groups
$(\Gamma _n, \Sigma _n)$
are pairwise isomorphic, and in particular isomorphic to
$(\Gamma _\infty , \Sigma _\infty )$
. We set
$G=\Gamma _\infty $
,
$\varphi _\infty = \text {id}$
and
$\varphi _n'$
a fixed marked isomorphism between
$(\Gamma _\infty , \Sigma _\infty )$
and
$(\Gamma _n, \Sigma _n)$
. By Corollary 5.9 and Proposition 5.3 we can find
$s>0$
such that sys
$(\Gamma _n,X)\geq 2s$
for every n. By definition of equivariant pointed Gromov–Hausdorff convergence for each element
$g\in \Sigma _\infty $
there exists
$g_n\in \Gamma _n$
such that
$d(g_n x, gx) < s$
, if n is sufficiently large. By the condition on the systole we deduce that
$g_n$
is unique. Finally, by the definition of R, if n is taken possibly larger, every such
$g_n$
belongs to
$\Sigma _n$
. Clearly this correspondence
$g \mapsto g_n$
is one-to-one. This means that there exists a permutation
$\mathcal {P}_n$
of the set
$\Sigma _n$
such that
$\varphi _n = \mathcal {P}_n \circ \varphi _n'$
is again a marked isomorphism between
$(\Gamma _\infty , \Sigma _\infty )$
and
$(\Gamma _n,\Sigma _n)$
such that
$\varphi _n(g)=g_n$
. It is now trivial to show that
$\varphi _n(g)$
converges uniformly on compact subsets of X to g for every
$g\in \Sigma _\infty $
, and so that
$\varphi _n$
converges algebraically to
$\varphi _\infty $
.
We now show (ii). By Corollary 3.14 it is enough to show that
$\Gamma _\omega = \varphi _\infty (G)$
for every non-principal ultrafilter
$\omega $
, where
$\Gamma _\omega $
is the ultralimit group of the sequence
$\Gamma _n$
. Here we are using Lemma 3.9 to identify the ultralimit group of the sequence
$(X,x,\Gamma _n)$
with a group of isometries of X. By Proposition 3.13 we know that there is a subsequence
$\lbrace n_k \rbrace $
such that
$(X,x, \Gamma _{n_k}) \underset {\mathrm {eq-pGH}}{\longrightarrow }(X,x, \Gamma _\omega )$
because X is proper. Therefore by the first part of the theorem there exists a homomorphism
$\psi \colon G \to \text {Isom}(X)$
with
$\psi (G)=\Gamma _\omega $
and
$\varphi _{n_k}\underset {\mathrm {alg}}{\longrightarrow } \psi $
. So
$\psi = \varphi _\infty $
by the uniqueness of the algebraic limit. This shows that
$\Gamma _\omega = \varphi _\infty (G)$
and concludes the proof.
We observe that the first part of the argument above shows the following corollary.
Corollary 7.7. In the standard setting of convergence the groups
$\Gamma _n$
are eventually isomorphic to
$\Gamma _\infty $
.
8 Examples
We show that each assumption on the class
$\mathcal {M}(\delta ,D)$
is necessary in order to have the discreteness of the limit group.
Example 8.1. (Non-elementarity of the group)
We take
$X_n=\mathbb {R}$
,
$x_n=0$
and
$\Gamma _n = \mathbb {Z}_{{1}/{n}}$
, the group generated by the translation of length
${1}/{n}$
. It is easy to show that
$(X_\omega , x_\omega )=(\mathbb {R},0)$
and
$\Gamma _\omega $
is the group of all translations of
$\mathbb {R}$
, for every non-principal ultrafilter
$\omega $
. Clearly
$\Gamma _\omega $
, and therefore any possible limit under equivariant pointed Gromov–Hausdorff convergence, is not discrete. Observe that each
$X_n$
is a proper, geodesic,
$0$
-hyperbolic metric space and each
$\Gamma _n$
is discrete, torsion-free and cocompact with codiameter
$\leq {1}/{n}$
.
Example 8.2. (Non-uniform bound on the diameter)
For every n we take a hyperbolic surface of genus
$2$
with systole equal to
$ {1}/{n}$
. Its fundamental group acts cocompactly on
$X_n=\mathbb {H}^2$
as a subgroup
$\Gamma _n$
of isometries. We take a basepoint
$x_n \in \mathbb {H}^2$
which belongs to the axis of an isometry of
$\Gamma _n$
with translation length
${1}/{n}$
. As in the example above
$\Gamma _\omega $
contains all the possible translations along an axis of
$X_\omega = \mathbb {H}^2$
and so it is not discrete, for every non-principal ultrafilter
$\omega $
. Observe that each
$X_n$
is a proper, geodesic,
$\log 3$
-hyperbolic metric space and each
$\Gamma _n$
is discrete, non-elementary, torsion-free and cocompact. However, the codiameter of
$\Gamma _n$
is not uniformly bounded above.
Example 8.3. (Groups with torsion)
Let Y be the wedge of a hyperbolic surface S of genus
$2$
and a sphere
$\mathbb {S}^2$
and let X be its universal cover, which is Gromov-hyperbolic. Denote by
$G_n$
the group of isometries of Y generated by the isometry that fixes S and acts as a rotation of angle
${2\pi }/{n}$
on
$\mathbb {S}^2$
fixing the wedging point. Let
$\Gamma _n$
be the covering group of
$G_n$
acting on
$X_n := X$
by isometries. The action of
$\Gamma _n$
is clearly discrete with bounded codiameter. However,
$\Gamma _\omega $
is not discrete.
Example 8.4. (Gromov-hyperbolicity)
Let
$X=\mathbb {R}^2$
,
$x = (0,0)$
and
$\Gamma _n$
be the cocompact, discrete, torsion-free group generated by the translations of vectors
$({1}/{n},0)$
and
$(0,1)$
. It is clear that
$\Gamma _\omega $
is not discrete for every non-principal ultrafilter
$\omega $
.
