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Random walks and contracting elements II: Translation length and Quasi-isometric embedding

Inhyeok Choi

TL;DR

The paper develops a comprehensive framework for random walks on metric spaces with contracting elements, establishing that random subgroups generated by non-elementary walks are quasi-isometrically embedded into the ambient space. It introduces two complementary methods: deviation inequalities and Gouel-style pivoting, to prove exponential genericity of contracting elements, a CLT for translation length, and sublinear/sublinear-discrepancy bounds between displacement and translation length. It further shows that independent random walks yield quasi-isometric embeddings and even free subgroups with high probability, and provides effective, quantitative estimates that feed into a counting problem showing exponential genericity for contracting elements and their translation lengths. These results extend the theory of random subgroups beyond hyperbolic or WPD settings to a broad class of spaces with contracting elements, with implications for random extensions and convex-cocompact-type phenomena in diverse geometric groups.

Abstract

Continuing from a companion article: 'Random walks and contracting elements I: Deviation inequality and limit laws', we study random walks on metric spaces with contracting elements. We prove that random subgroups of the isometry group of a metric space is quasi-isometrically embedded into the space. We discuss this problem in two senses, namely, one involving random walks and the other involving counting problems. We also establish the genericity of contracting elements and the CLT and its converse for translation length.

Random walks and contracting elements II: Translation length and Quasi-isometric embedding

TL;DR

The paper develops a comprehensive framework for random walks on metric spaces with contracting elements, establishing that random subgroups generated by non-elementary walks are quasi-isometrically embedded into the ambient space. It introduces two complementary methods: deviation inequalities and Gouel-style pivoting, to prove exponential genericity of contracting elements, a CLT for translation length, and sublinear/sublinear-discrepancy bounds between displacement and translation length. It further shows that independent random walks yield quasi-isometric embeddings and even free subgroups with high probability, and provides effective, quantitative estimates that feed into a counting problem showing exponential genericity for contracting elements and their translation lengths. These results extend the theory of random subgroups beyond hyperbolic or WPD settings to a broad class of spaces with contracting elements, with implications for random extensions and convex-cocompact-type phenomena in diverse geometric groups.

Abstract

Continuing from a companion article: 'Random walks and contracting elements I: Deviation inequality and limit laws', we study random walks on metric spaces with contracting elements. We prove that random subgroups of the isometry group of a metric space is quasi-isometrically embedded into the space. We discuss this problem in two senses, namely, one involving random walks and the other involving counting problems. We also establish the genericity of contracting elements and the CLT and its converse for translation length.
Paper Structure (15 sections, 6 theorems, 195 equations)

This paper contains 15 sections, 6 theorems, 195 equations.

Table of Contents

  1. Introduction
  2. Structure of the paper
  3. Preliminaries
  4. Contracting isometries and alignment
  5. Random walks
  6. Schottky set
  7. The first method: deviation inequalities
  8. The second method: pivoting technique
  9. Pivotal times and pivoting
  10. Reduction for random walks
  11. Pivoting and self-repulsion
  12. Converse of CLT
  13. Pivoting and repulsion among independent random walks
  14. Effective estimates for translation length
  15. Counting problem

Key Result

Theorem A

Let $(X, G, o)$ be as in Convention conv:main and let $(Z_{n})_{n \ge 0}$ be the random walk generated by a non-elementary measure $\mu$ on $G$. Let $\lambda(\mu)$ be the escape rate of $\mu$, i.e., $\lambda(\mu) := \lim_{n \rightarrow \infty} \mathop{\mathrm{\mathbb{E}}}\nolimits_{\mu^{\ast n}} [d(

Theorems & Definitions (36)

  • Theorem A
  • Theorem B
  • Theorem C: CLT and its converse
  • Theorem D
  • Theorem E
  • Definition 2.1: contracting sets
  • Definition 2.2: Translation length
  • Definition 2.3: choi2022random1
  • Definition 2.8: cf. gouezel2022exponential, choi2022random1
  • Definition 2.10
  • ...and 26 more