Topological full groups of line-like minimal group actions are amenable

TL;DR

This paper proves that the topological full group of a finitely generated minimal group action on a compact space, with a Schreier graph quasi-isometric to a line, is amenable.

Contribution

It establishes the amenability of topological full groups under specific geometric conditions on the action's Schreier graph.

Findings

Topological full groups are amenable when the Schreier graph is line-like.

The result applies to finitely generated minimal actions on compact spaces.

Provides new connections between geometric group theory and topological dynamics.

Abstract

We consider a finitely generated group acting minimally on a compact space by homeomorphsims, and assume that the Schreier graph of at least one orbit is quasi-isometric to a line. We show that the topological full group of such an action is amenable.