Two-ended quasi-transitive graphs.

The well-known characterization of two-ended groups says that every two-ended group can be split over finite subgroups which means it is isomorphic to either by a free product with amalgamation A * C B or an HNN-extension * ϕ C , where C is a finite group and [ A : C ] = [ B : C ] = 2 and ϕ ∈ Aut (C). In this paper, we show that there is a way in order to spilt two-ended quasi-transitive graphs without dominated ends and two-ended transitive graphs over finite subgraphs in the above sense. As an application of it, we characterize all groups acting with finitely many orbits almost freely on those graphs. [ABSTRACT FROM AUTHOR]