This lecture is a presentation of the preprint https://arxiv.org/abs/1804.02995. It will concern discrete invariant random subgroups in isometry groups of Gromov-hyperbolic spaces. In the case of rank one Lie groups essentially all IRSs are known to be discrete, but in general this is a nontrivial assumption.

Critical exponents

Let \( X \) be a Gromov-hyperbolic space and \( o \in X \). Let \( \Gamma \) be a discrete subgroup of \( \mathrm{Isom}(X) \). For any \( s \ge 0 \) we can define the Poincaré series
\[
P(\Gamma, s) = \sum_{\gamma \in \Gamma} e^{-sd(o, \gamma o)} \in [0, +\infty].
\]
The critical exponent \( \delta(\Gamma) \) of \( \Gamma \) is then defined by
\[
\delta(\Gamma) = \inf(s > 0 :\: P(\Gamma, s) < +\infty) = \sup(s \ge 0 :\: P(\Gamma, s) = +\infty).
\]
It measures the exponential growth rate of the orbits of \( \Gamma \) on \( X \), as it is easily seen to also be given by the following formula:
\[
\delta(\Gamma) = \liminf_{R \to +\infty} \left( \frac{\log|B(o, R) \cap \Gamma\cdot o|}{R} \right).
\]
It is also readily seen that it does not depend on the chosen origin \( o \).

Part of the finer asymptotics of this growth is captured by the following notion: if \( P(\Gamma, s) < +\infty \) for \( s = \delta(\Gamma) \) then \( \Gamma \) is said to be of convergence type; otherwise it is said to be of divergence type.

This is a well-studied invariant and a sample of its known properties is given by the following list:

• \( \delta(\Gamma) = 0 \) if and only if \( \Gamma \) is elementary;
• Let \( \partial X \) be the Gromov boundary of \( X \); then if \( \Gamma \) is a uniform lattice (acts cocompactly on \( X \)) we have \( \delta(\Gamma) = \dim(\partial X) \) (where \( \dim \) is the Hausdorff dimension of a metric space);
• For any discrete subgroup \( \Gamma \le \mathrm{Isom}(X) \) we have \( 0 \le \delta(\Gamma) \le \dim(\partial X) \). Moreover there are subgroups with \( \delta(\Gamma) \) arbitrarily close to zero, which may be taken to be of either convergenec or divergence type;
• \( \delta(\Gamma) \) equals the dimension of the radial, or conical, limt set of \( \Gamma \).

Main result

Viewed as a function on Chabauty space (or rather its subspace containing only discrete groups) \( \delta \) is not continuous. An important property for the sequel is the following weaker property.

The function \( \delta \) is measurable on the discrete part of the Chabauty space.

As it is obvious that \( \delta \) is conjugacy-invariant, it follows that if \( \mu \) is an ergodic IRS of \( \mathrm{Isom}(X) \) the value \( \delta(\Gamma) \) is \( \mu \)-essentially constant and we can take its value to be \( \delta(\mu) \). Similarly, being of divergence/convergence type is a Borel property and hence well-defined for ergodic IRSs.

Theorem (Gekhtman–Levit): Let \( \mu \) be an ergodic discrete IRS of \( \mathrm{Isom}(X) \). Then
\[
\delta(\mu) \ge \frac 1 2 \dim(\partial X).
\]
Moreover, if \( \mu \) is of divergence type then necessarily \( \delta(\Gamma) = \dim(\partial X) \).

The main tool in the proof is an ergodic theorem due to Nevo and Zimmer.

Consequences

A theorem of Corlette states that if \( G \) is a Lie group of rank 1 with Kazhdan’s property (T) then \( \delta(\Gamma) = \dim(X) \) if and only if \( \Gamma \) is a lattice in \( G \). Thus, a corollary of the theorem above is that

If \( G \) is as above then any ergodic IRS of \( G \) of divergence type is supported on the conjugacy class of a lattice in \( G \).