Vertex separators, chordality and virtually free groups

TL;DR

This paper explores the use of minimal vertex separators and generalized chordality to characterize virtually free groups and analyze properties of Baumslag-Solitar groups, with applications to the word problem.

Contribution

It introduces new characterizations of virtually free groups using graph-theoretic tools and applies generalized chordality to study Baumslag-Solitar groups and the word problem.

Findings

Two new characterizations of virtually free groups.

Baumslag-Solitar group BS(1,n) is k-chordal iff |n|<3.

Application of generalized chordality to the word problem.

Abstract

In this paper we consider some results obtained for graphs using minimal vertex separators and generalized chordality and translate them to the context of Geometric Group Theory. Using these new tools, we are able to give two new characterizations for a group to be virtually free. Furthermore, we prove that the Baumslag-Solitar group $BS(1,n)$ is $k$-chordal for some $k$ if and only if $|n|<3$ and we give an application of generalized chordality to the study of the word problem.