A Stallings' type theorem for quasi-transitive graphs

Abstract

We consider infinite connected quasi-transitive locally finite graphs and show that every such graph with more than one end is a tree amalgamation of two other such graphs. This can be seen as a graph-theoretical version of Stallings' splitting theorem for multi-ended finitely generated groups and indeed it implies this theorem. It will also lead to a characterisation of accessible graphs. We obtain applications of our results for hyperbolic graphs, planar graphs and graphs without any thick end. The application for planar graphs answers a question of Mohar in the affirmative.

Minor exclusion in quasi-transitive graphs

“…According to Krön and Möller [6, Theorem 2.8], every end of G is thin. Thus, [4,Theorem 7.5] and [3, Lemma 2.9] imply that G is quasi-isometric to a 3-regular tree. By Proposition 2.3, G is minor excluded.…”

Two characterisations of accessible quasi-transitive graphs

“…So far, the tree amalgamation do not interact with any group action. In the following, we define some notions that ensure that tree amalgamations of quasi-transitive graphs that satisfy this notion are again quasi-transitive, see [4,Lemma 5.3].…”

“…The question arises which quasi-transitive locally finite connected graphs are accessible. A result of [4] says that such graphs are accessible if and only if they are accessible in the sense of Thomassen and Woess [7]. (We refer to Section 2 for their definition of accessibility.)…”

“…It is clear from the definition that some process of splittings stops if and only if the graph has a terminal factorisation and thus is accessible. It was conjectured in [4] that the property of stopping of the process of splittings does not depend on the particular splittings. To be precise, it was conjectured that one process of splittings stops after finitely many steps if and only if every process does this.…”

Splitting groups with cubic Cayley graphs of connectivity two

“…Lemma 2.2. [12,Corollary 4.3] Let Γ be a locally finite graph with more than one end such that a group G acts on Γ. Then there exists a tree-decomposition (T, V) with the following properties:…”