Computational complexity and the conjugacy problem

TL;DR

This paper constructs finitely generated groups with highly variable conjugacy problem complexities, demonstrating that the problem can range from quadratic time decidability to arbitrary c.e. degrees, and explores related complexity phenomena.

Contribution

It introduces groups with customizable conjugacy problem complexities, including arbitrary Turing degrees and generic linear time solutions, advancing understanding of computational group theory.

Findings

Conjugacy problem complexity can be tailored to any c.e. Turing degree.

Existence of groups with generically linear time conjugacy algorithms.

Discussion of groups with algorithmically finite conjugation.

Abstract

The conjugacy problem for a finitely generated group $G$ is the two-variable problem of deciding for an arbitrary pair $(u,v)$ of elements of $G$, whether or not $u$ is conjugate to $v$ in $G$. We construct examples of finitely generated, computably presented groups such that for every element $u_0$ of $G$, the problem of deciding if an arbitrary element is conjugate to $u_0$ is decidable in quadratic time but the worst-case complexity of the global conjugacy problem is arbitrary: it can be any c.e. Turing degree , can exactly mirror the Time Hierarchy Theorem, or can be $\mathcal{NP}$-complete. Our groups also have the property that the conjugacy problem is generically linear time: that is, there is a linear time partial algorithm for the conjugacy problem whose domain has density $1$, so hard instances are very rare. We also consider the complexity relationship of the "half-conjugacy" problem to the conjugacy problem. In the last section we discuss the extreme opposite situation: groups with algorithmically finite conjugation.