Improved Parallel Algorithms for Baumslag Groups

TL;DR

This paper improves the complexity bounds for solving the word and conjugacy problems in Baumslag groups using refined power circuit algorithms, achieving results in uniform TC^1 and related classes.

Contribution

It refines power circuit operations to significantly improve the complexity of key problems in Baumslag groups, including new upper bounds in uniform TC^1.

Findings

Word problem in G_{p,pq} solvable in uTC^1

Conjugacy problem in G_{p,pq} generically in uTC^1

Conjugacy to fixed element in G_{1,q} decidable in uTC^1

Abstract

The Baumslag group had been a candidate for a group with an extremely difficult word problem until Myasnikov, Ushakov, and Won succeeded to show that its word problem can be solved in polynomial time. Their result used the newly developed data structure of power circuits allowing for a non-elementary compression of integers. Later this was extended in two directions: Laun showed that the same applies to the Baumslag groups $G_{1, q}$ for $q \geq 2$ and we established that the word problem of the Baumslag group $G_{1, 2}$ can be solved in $\mathsf{TC}^2$. In this work we refine the operations on reduced power circuits to further improve upon both results. We show that the word problem of the Baumslag groups $G_{p, pq}$ with $|p|,|q| \geq 1$ can be solved in $\mathsf{uTC}^1$. Moreover, we prove that the conjugacy problem in $G_{p, pq}$ is strongly generically in $\mathsf{uTC}^1$ (meaning that for "most" inputs it is in $\mathsf{uTC}^1$). Finally, for every fixed $g \in G_{1, q}$ (case $p=1$) conjugacy to $g$ can be decided in $\mathsf{uTC}^1$ for all inputs. We further show that the word problem of the Baumslag-Solitar groups $BS_{p, pq}$ is in $\mathsf{uAC}^0(F_2)$ if the input word is given in a quite compressed form and so give a complexity result for a special case of the power word problem for these groups.