The $R_{\infty}$ property for Houghton's groups

TL;DR

This paper proves that all Houghton's groups, which are groups of translations on multiple rays of points at infinity, possess the $R_{ty}$ property, indicating infinite twisted conjugacy classes for all automorphisms.

Contribution

It establishes that every Houghton's group $ ext{H}_n$ has the $R_{ty}$ property, a significant result in understanding their automorphism structure.

Findings

All Houghton's groups $ ext{H}_n$ have the $R_{ty}$ property.

The $R_{ty}$ property holds for all $n eq 0$ in the family of Houghton's groups.

This advances the understanding of twisted conjugacy classes in infinite groups.

Abstract

We study twisted conjugacy classes of a family of groups which are called Houghton's groups $\mathcal{H}_n$ ($n \in\mathbb{N}$), the group of translations of $n$ rays of discrete points at infinity. We prove that the Houghton's groups $\mathcal{H}_n$ have the $R_{\infty}$ property for all $n\in \mathbb{N}$.