The conjugacy problem in free solvable groups and wreath product of abelian groups is in TC$^0$

TL;DR

This paper demonstrates that the conjugacy problem in free solvable groups and wreath products of abelian groups can be efficiently solved within the complexity class TC$^0$, by reducing it to simpler problems in the factors.

Contribution

It establishes a uniform TC$^0$ complexity reduction of the conjugacy problem in wreath products and free solvable groups to problems in their factors, improving understanding of their computational complexity.

Findings

Conjugacy problem in wreath products reduces to factor problems in TC$^0$.

In free solvable groups, the conjugacy problem is in TC$^0$ due to reductions.

The results apply to iterated wreath products of abelian groups and free solvable groups.

Abstract

We show that the conjugacy problem in a wreath product $A \wr B$ is uniform-$\mathsf{TC}^0$-Turing-reducible to the conjugacy problem in the factors $A$ and $B$ and the power problem in $B$. If $B$ is torsion free, the power problem for $B$ can be replaced by the slightly weaker cyclic submonoid membership problem for $B$. Moreover, if $A$ is abelian, the cyclic subgroup membership problem suffices, which itself is uniform-$\mathsf{AC}^0$-many-one-reducible to the conjugacy problem in $A \wr B$. Furthermore, under certain natural conditions, we give a uniform $\mathsf{TC}^0$ Turing reduction from the power problem in $A \wr B$ to the power problems of $A$ and $B$. Together with our first result, this yields a uniform $\mathsf{TC}^0$ solution to the conjugacy problem in iterated wreath products of abelian groups - and, by the Magnus embedding, also in free solvable groups.