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Examples of measures with trivial left and non-trivial right random walk tail boundary

Andrei Alpeev

Abstract

In early 80's Vadim Kaimanovich presented a construction of a non-degenerate measure, on the standard lamplighter group, that has a trivial left and non-trivial right random walk tail boundary. We show that examples of such kind are possible precisely for amenable groups that have non-trivial factors with infinite conjugacy classes property.

Examples of measures with trivial left and non-trivial right random walk tail boundary

Abstract

In early 80's Vadim Kaimanovich presented a construction of a non-degenerate measure, on the standard lamplighter group, that has a trivial left and non-trivial right random walk tail boundary. We show that examples of such kind are possible precisely for amenable groups that have non-trivial factors with infinite conjugacy classes property.

Paper Structure

This paper contains 4 sections, 15 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. The maximal ICC factor
  3. A process with heavy tail
  4. Construction

Key Result

Theorem 1

Let $G$ be a countable group. There is a non-degenerate probability measure $\nu$ on $G$ with trivial left and non-trivial right random walk tail boundaries iff $G$ is amenable and has a non-trivial ICC factor-group. Moreover, the action of maximal ICC factor on the boundary is essentially free.

Theorems & Definitions (23)

  • Theorem 1
  • Proposition 1
  • proof
  • Lemma 1
  • Theorem 2
  • Lemma 2
  • Remark 1
  • Lemma 3
  • Lemma 4
  • Proposition 2
  • ...and 13 more