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

TL;DR

This paper investigates the behavior of random walks on metric spaces with contracting elements, demonstrating quasi-isometric embeddings of random subgroups, establishing genericity of contracting elements, and analyzing translation length with CLT results.

Contribution

It extends previous work by proving quasi-isometric embeddings of random subgroups and analyzing the distribution of translation lengths in this setting.

Findings

Random subgroups are quasi-isometrically embedded into the metric space.

Contracting elements are shown to be generic among the elements considered.

A Central Limit Theorem (CLT) for translation length is established, along with its converse.

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.