Finite automata for Schreier graphs of virtually free groups

TL;DR

This paper generalizes Stallings' construction for free groups to virtually free groups using Stallings sections, enabling efficient computation of Schreier graph cores and exploring related complexity and applications.

Contribution

It introduces Stallings sections to characterize virtually free groups and provides a constructive approach based on Bass-Serre theory for core computation.

Findings

Groups with Stallings sections are exactly finitely generated virtually free groups

Efficient algorithms for core computation of Schreier graphs are developed

Discusses complexity and potential applications of the method

Abstract

The Stallings construction for finitely generated subgroups of free groups is generalized by introducing the concept of Stallings section, which allows an eficient computation of the core of a Schreier graph based on edge folding. It is proved that those groups admitting Stallings sections are precisely finitely generated virtually free groups, through a constructive approach based on Bass-Serre theory. Complexity issues and applications are also discussed.