Conjugacy in Houghton's Groups

TL;DR

This paper proves that the conjugacy and conjugator search problems are solvable in Houghton's groups, which are groups of permutations acting as translations on multiple copies of natural numbers.

Contribution

It establishes the solvability of the conjugacy problem and conjugator search problem specifically for Houghton's groups, advancing understanding of their algorithmic properties.

Findings

Conjugacy problem is solvable in Houghton's groups.

Conjugator search problem is solvable in Houghton's groups.

Provides algorithms for solving these problems in $H_n$, n ≥ 2.

Abstract

Let $n\in \mathbb{N}$. Houghton's group $H_n$ is the group of permutations of $\{1,\dots, n\}\times \mathbb{N}$, that eventually act as a translation in each copy of $\mathbb{N}$. We prove the solvability of the conjugacy problem and conjugator search problem for $H_n$, $n\geq 2$.