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

TL;DR

This paper explores specific measures on groups that exhibit trivial left tail boundaries but non-trivial right tail boundaries, revealing their existence is tied to certain properties of amenable groups with infinite conjugacy classes.

Contribution

It characterizes the class of groups where such measures can exist, linking the phenomenon to amenability and the infinite conjugacy classes property.

Findings

Examples exist only for amenable groups with non-trivial factors having infinite conjugacy classes.

The construction generalizes Kaimanovich's 1980s example on lamplighter groups.

Provides a classification of groups supporting measures with trivial left and non-trivial right tail boundaries.

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.