Evacuation schemes on Cayley graphs and non-amenability of groups

TL;DR

This paper introduces evacuation schemes on Cayley graphs, linking their existence to non-amenability of groups, and explores their properties and implications for Thompson's group F, an open problem in group theory.

Contribution

It defines evacuation schemes on Cayley graphs, establishes their connection to non-amenability, and analyzes their existence in relation to Thompson's group F.

Findings

Evacuation schemes are equivalent to non-amenability of the group.

Pure evacuation schemes do not exist for certain generators of Thompson's group F.

Existence of schemes with edges used twice remains an open question.

Abstract

In this paper we introduce a concept of an evacuation scheme on the Cayley graph of an infinite finitely generated group. This is a collection of infinite simple paths bringing all vertices to infinity. We impose a restriction that every edge can be used a uniformly bounded number of times in this scheme. An easy observation shows that existing of such a scheme is equivalent to non-amenability of the group. A special case happens if every edge can be used only once. These scheme are called pure. We obtain a criterion for existing of such a scheme in terms of isoperimetric constant of the graph. We analyze R.\,Thompson's group $F$, for which the amenability property is a famous open problem. We show that pure evacuation schemes do not exist for the set of generators $\{x_0,x_1,\bar{x}_1\}$, where $\bar{x}_1=x_1x_0^{-1}$. However, the question becomes open if edges with labels $x_0^{\pm1}$ can be used twice. Existing of pure evacuation scheme for this version is implied by some natural conjectures.