Evacuation schemes on Cayley graphs and non-amenability of groups

In this paper, we introduce a concept of an evacuation scheme on the Cayley graph infinite finitely generated group. This is collection simple paths bringing all vertices to infinity. We impose restriction that every edge can be used uniformly bounded number times in scheme. An easy observation shows existence such equivalent non-amenability A special case happens if only once. These schemes are called pure. obtain criterion for terms isoperimetric constant graph. analyze R. Thompson’s group [Formula: see text], which amenability property famous open problem. show pure do not exist set generators where text]. However, question becomes edges with labels text] twice. The version implied by some natural conjectures.