Evacuation schemes on Cayley graphs and non-amenability of groups

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 the existence 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 schemes are called pure. We obtain a criterion for the existence of such a scheme in terms of isoperimetric constant of the graph. We analyze R. Thompson’s group [Formula: see text], for which the amenability property is a famous open problem. We show that pure evacuation schemes do not exist for the set of generators [Formula: see text], where [Formula: see text]. However, the question becomes open if edges with labels [Formula: see text] can be used twice. The existence of pure evacuation schemes for this version is implied by some natural conjectures.