A characterisation of virtually free groups via minor exclusion

TL;DR

This paper characterizes virtually free groups through the concept of graph minors, establishing a new criterion based on Cayley graphs being minor excluded for any finite generating set.

Contribution

It provides a novel graph-theoretic characterization of virtually free groups, answering a previously open question and linking group properties to minor exclusion in Cayley graphs.

Findings

Virtually free groups are characterized by minor exclusion in all Cayley graphs.

Finitely generated, infinite groups with minor excluded Cayley graphs are accessible.

The proof connects graph minor theory with group accessibility concepts.

Abstract

We give a new characterisation of virtually free groups using graph minors. Namely, we prove that a finitely generated, infinite group is virtually free if and only if for any finite generating set, the corresponding Cayley graph is minor excluded. This answers a question of Ostrovskii and Rosenthal. The proof relies on showing that a finitely generated group that is minor excluded with respect to every finite generating set is accessible, using a graph-theoretic characterisation of accessibility due to Thomassen and Woess.