Amenability of trees

TL;DR

This paper establishes a criterion for determining when locally finite trees are amenable, using a novel trimming operator, and applies it to characterize amenability in Galton-Watson trees.

Contribution

It introduces a new criterion based on a trimming operator for assessing amenability of trees, including Galton-Watson trees.

Findings

Criterion for amenability of locally finite trees

Necessary and sufficient condition for Galton-Watson trees

Introduction of a trimming operator for trees

Abstract

We will give a criterion for the amenability of arbitrary locally finite trees. The criterion is based on the trimming operator which is defined on the space of trees. As an application, we obtain a necessary and sufficient condition for that amenability of Galton-Watson trees.