Geometric amenability in totally disconnected locally compact groups

Abstract We give a short geometric proof of result Soardi and Woess Salvatori that quasitransitive graph is amenable if only its automorphism group unimodular. also strengthen one direction by showing compactly generated totally disconnected locally compact admits proper Lipschitz action on bounded degree then pass via the notion amenability group, which has previously been studied second author defined analogy with amenability, using right Følner sets instead left sets. introduce uniform non-amenability relate this in various ways to actions graphs modular homomorphism.