Sigma-compactness of Morse boundaries in Morse local-to-global groups and applications to stationary measures

TL;DR

This paper proves the sigma-compactness of Morse boundaries in Morse local-to-global groups, explores the converse for small cancellation groups, and shows that certain Morse boundaries do not admit non-trivial stationary measures, without extra assumptions.

Contribution

It establishes sigma-compactness of Morse boundaries in Morse local-to-global groups and characterizes stationary measures on their boundaries, extending previous results without additional assumptions.

Findings

Morse boundary of Morse local-to-global groups is sigma-compact.

Converse holds for small cancellation groups.

Stationary measures on certain Morse boundaries are trivial.

Abstract

We show that the Morse boundary of a Morse local-to-global group is $\sigma$-compact. Moreover, we show that the converse holds for small cancellation groups. As an application, we show that the Morse boundary of a non-hyperbolic, Morse local-to-global group that has contraction does not admit a non-trivial stationary measure. In fact, we show that any stationary measure on a geodesic boundary of such a groups needs to assign measure zero to the Morse boundary. Unlike previous results, we do not need any assumptions on the stationary measures considered.