The isomorphism problem for finite extensions of free groups is in PSPACE

TL;DR

This paper introduces an algorithm to solve the isomorphism problem for finite extensions of free groups within PSPACE, utilizing graph of groups constructions and complexity analysis.

Contribution

It provides a new algorithmic approach to the isomorphism problem for virtually free groups with complexity bounds, improving understanding of computational aspects.

Findings

Algorithm for constructing graph of groups from context-free grammars

Isomorphism problem solvable in doubly exponential space

Construction complexity reduces to NP for finite extensions of free groups

Abstract

We present an algorithm for the following problem: given a context-free grammar for the word problem of a virtually free group $G$, compute a finite graph of groups $\mathcal{G}$ with finite vertex groups and fundamental group $G$. Our algorithm is non-deterministic and runs in doubly exponential time. It follows that the isomorphism problem of context-free groups can be solved in doubly exponential space. Moreover, if, instead of a grammar, a finite extension of a free group is given as input, the construction of the graph of groups is in NP and, consequently, the isomorphism problem in PSPACE.