# 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.

## Key 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$^0$.