The power word problem

TL;DR

This paper introduces the power word problem, a new variant of the word problem in finitely generated groups, analyzing its complexity across different group classes and establishing reductions and hardness results.

Contribution

It defines the power word problem, relates it to the compressed word problem, and determines its complexity for various groups, including free groups, wreath products, and abelian groups.

Findings

Power word problem for free groups reduces to the word problem in AC^0.

Power word problem for certain wreath products is coNP-complete.

Power word problem for abelian groups is in TC^0.

Abstract

In this work we introduce a new succinct variant of the word problem in a finitely generated group $G$, which we call the power word problem: the input word may contain powers $p^x$, where $p$ is a finite word over generators of $G$ and $x$ is a binary encoded integer. The power word problem is a restriction of the compressed word problem, where the input word is represented by a straight-line program (i.e., an algebraic circuit over $G$). The main result of the paper states that the power word problem for a finitely generated free group $F$ is AC$^0$-Turing-reducible to the word problem for $F$. Moreover, the following hardness result is shown: For a wreath product $G \wr \mathbb{Z}$, where $G$ is either free of rank at least two or finite non-solvable, the power word problem is complete for coNP. This contrasts with the situation where $G$ is abelian: then the power word problem is shown to be in TC$^0$.

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