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Stationary random subgroups in negative curvature

Ilya Gekhtman, Arie Levit

TL;DR

This work develops a probabilistic framework for stationary random subgroups in negative curvature spaces, linking limit-set behavior and critical exponents via random walks and boundary measures. It establishes that discrete $\mu$-stationary subgroups not contained in the elliptic radical have full limit sets and exhibit lower bounds on their critical exponents, with sharper results for divergence-type subgroups. The paper then translates these geometric-analytic bounds into non-confinement results, showing that subgroups with sufficiently large growth cannot be confined, and extends the theory to products and CAT$(0)$ contexts. These results yield implications for spectral data of locally symmetric spaces and provide a versatile method to connect random-walk entropy, drift, and subgroup geometry in rank-one and CAT$(-1)$ settings. The approach leverages Kakutani's ergodic theorem, boundary hitting measures, Tanaka-type arguments, and Chabauty-continuity of the critical exponent to unify limit-set, density, and confinement phenomena for stationary random subgroups.

Abstract

We show that discrete stationary random subgroups of isometry groups of Gromov hyperbolic spaces have full limit sets as well as critical exponents bounded from below. This information is used to answer a question of Gelander and show that a rank one locally symmetric space for which the bottom of the spectrum of the Laplace-Beltrami operator is the same as that of its universal cover has unbounded injectivity radius.

Stationary random subgroups in negative curvature

TL;DR

This work develops a probabilistic framework for stationary random subgroups in negative curvature spaces, linking limit-set behavior and critical exponents via random walks and boundary measures. It establishes that discrete -stationary subgroups not contained in the elliptic radical have full limit sets and exhibit lower bounds on their critical exponents, with sharper results for divergence-type subgroups. The paper then translates these geometric-analytic bounds into non-confinement results, showing that subgroups with sufficiently large growth cannot be confined, and extends the theory to products and CAT contexts. These results yield implications for spectral data of locally symmetric spaces and provide a versatile method to connect random-walk entropy, drift, and subgroup geometry in rank-one and CAT settings. The approach leverages Kakutani's ergodic theorem, boundary hitting measures, Tanaka-type arguments, and Chabauty-continuity of the critical exponent to unify limit-set, density, and confinement phenomena for stationary random subgroups.

Abstract

We show that discrete stationary random subgroups of isometry groups of Gromov hyperbolic spaces have full limit sets as well as critical exponents bounded from below. This information is used to answer a question of Gelander and show that a rank one locally symmetric space for which the bottom of the spectrum of the Laplace-Beltrami operator is the same as that of its universal cover has unbounded injectivity radius.
Paper Structure (10 sections, 62 theorems, 145 equations)

This paper contains 10 sections, 62 theorems, 145 equations.

Table of Contents

  1. Introduction
  2. Stationary measures and Kakutani's ergodic theorem
  3. Group actions on hyperbolic spaces
  4. Stationary closed subsets
  5. Geometric density for discrete stationary random subgroups
  6. Discrete stationary random subgroups of products
  7. $\mathrm{CAT}(0)$-spaces and geometric density
  8. Random walks and Poincare series
  9. Critical exponents of stationary random subgroups
  10. Random walks and confined subgroups

Key Result

Theorem 1.1

Let $\Gamma$ be a discrete subgroup of the simple rank one Lie group $G$. If the subgroup $\Gamma$ satisfies $\delta(\Gamma) \le \frac{1}{2}d(X)$ then $\Gamma$ is not confined.

Theorems & Definitions (119)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5: Fraczyk
  • Theorem 1.6
  • Theorem 1.7
  • Proposition 1.8
  • Theorem 1.9
  • Remark
  • ...and 109 more