A characterisation of virtually free groups

TL;DR

This paper characterizes virtually free groups by showing they can be identified through a specific property of locally geodesic words relative to some generating set, providing a new criterion for their recognition.

Contribution

It introduces a novel characterization of virtually free groups based on the behavior of k-locally geodesic words, linking local and global geodesic properties.

Findings

Virtually free groups can be characterized by local geodesic properties.

Existence of a generating set and k > 0 such that all k-locally geodesic words are geodesic.

Provides a new criterion for identifying virtually free groups.

Abstract

We prove that a finitely generated group $G$ is virtually free if and only if there exists a generating set for $G$ and $k > 0$ such that all $k$-locally geodesic words with respect to that generating set are geodesic.