Amenable group actions on infinite graphs (Q1105141)

Let \(\Gamma\) be a group of automorphisms of an arbitrary graph G which is infinite, locally finite and connected. We study the interplay between amenability of the action of \(\Gamma\) on G and the way in which \(\Gamma\) acts on the space of ends of G. If \(\Gamma\) acts amenably, then it fixes a finite set of vertices, an end or a pair of ends of G. Conversely, if \(\Gamma\) fixes a finite set of vertices, an end of finite diameter or a pair of ends, then its action on G is amenable. These results extend Nebbia's (1988) characterization of amenable automorphism groups of an infinite tree, which was preceded by a theorem of Tits (1977) on solvable groups of automorphisms of trees.