A fast algorithm for Stallings foldings over virtually free groups

TL;DR

This paper presents a simple, efficient algorithm for solving the subgroup membership problem in virtually free groups, with practical applications demonstrated in matrix groups like SL(2,Z).

Contribution

It introduces a new algorithm that solves the subgroup membership problem in virtually free groups in near-linear time, improving computational efficiency.

Findings

Algorithm solves subgroup membership in O(n log* n) time

Practical applications in SL(2,Z) and GL(2,Z)

Decides subgroup isomorphism to free groups

Abstract

We give a simple algorithm to solve the subgroup membership problem for virtually free groups. For a fixed virtually free group with a fixed generating set $X$, the subgroup membership problem is uniformly solvable in time $O(n\log^*(n))$ where $n$ is the sum of the word lengths of the inputs with respect to $X$. For practical purposes, this can be considered to be linear time. The algorithm itself is simple and concrete examples are given to show how it can be used for computations in $\mathrm{SL}(2,\mathbb Z)$ and $\mathrm{GL}(2,\mathbb Z)$. We also give an algorithm to decide whether a finitely generated subgroup is isomorphic to a free group.