Folner sets of alternate directed groups

TL;DR

This paper constructs explicit Folner sets for certain directed groups acting on rooted trees, offering a new proof of their amenability without relying on probabilistic methods or random walks.

Contribution

It introduces an explicit construction of Folner sets for directed groups with sublogarithmic valency, providing an alternative proof of amenability.

Findings

Explicit Folner sets are constructed for these groups.

The proof of amenability is independent of probabilistic methods.

Applicable to groups acting on rooted trees with sublogarithmic valency.

Abstract

An explicit family of Folner sets is constructed for some directed groups acting on a rooted tree of sublogarithmic valency by alternate permutations. In the case of bounded valency, these groups were known to be amenable by probabilistic methods. The present construction provides a new and independent proof of amenability, using neither random walks, nor word length.