A Logspace Solution to the Word and Conjugacy problem of Generalized Baumslag-Solitar Groups

TL;DR

This paper presents a LOGSPACE algorithm for solving the word and conjugacy problems in generalized Baumslag-Solitar groups, improving understanding of their computational complexity.

Contribution

It introduces a uniform LOGSPACE solution for both problems and analyzes the complexity of the uniform conjugacy problem.

Findings

Word problem reduces to free group in TC^0.

Conjugacy problem solvable in LOGSPACE for fixed groups.

Uniform conjugacy problem is EXPSPACE-complete.

Abstract

Baumslag-Solitar groups were introduced in 1962 by Baumslag and Solitar as examples for finitely presented non-Hopfian two-generator groups. Since then, they served as examples for a wide range of purposes. As Baumslag-Solitar groups are HNN extensions, there is a natural generalization in terms of graph of groups. Concerning algorithmic aspects of generalized Baumslag-Solitar groups, several decidability results are known. Indeed, a straightforward application of standard algorithms leads to a polynomial time solution of the word problem (the question whether some word over the generators represents the identity of the group). The conjugacy problem (the question whether two given words represent conjugate group elements) is more complicated; still decidability has been established by Anshel and Stebe for ordinary Baumslag-Solitar groups and for generalized Baumslag-Solitar groups independently by Lockhart and Beeker. However, up to now no precise complexity estimates have been given. In this work, we give a LOGSPACE algorithm for both problems. More precisely, we describe a uniform TC^0 many-one reduction of the word problem to the word problem of the free group. Then we refine the known techniques for the conjugacy problem and show that it can be solved in LOGSPACE. Moreover, for ordinary Baumslag-Solitar groups also conjugacy is AC^0-Turing-reducible to the word problem of the free group. Finally, we consider uniform versions (where also the graph of groups is part of the input) of both word and conjugacy problem: while the word problem still is solvable in LOGSPACE, the conjugacy problem becomes EXPSPACE-complete.