Growth rate of endomorphisms of Houghton's groups

TL;DR

This paper investigates the growth rates of endomorphisms of Houghton's groups, revealing that for non-trivial kernels the growth rate is either 1 or the spectral radius, and for monomorphisms it equals a specific translation length.

Contribution

It characterizes the growth rates of endomorphisms and monomorphisms of Houghton's groups, establishing a precise relationship with translation lengths and spectral radii.

Findings

Growth rate equals 1 or spectral radius for non-trivial kernels.

Monomorphisms have growth rate equal to a specific translation length.

Each monomorphism determines a unique translation length.

Abstract

A Houghton's group $\mathcal{H}_n$ consists of translations at infinity of a $n$ rays of discrete points on the plane. In this paper we study the growth rate of endomorphisms of Houghton's groups. We show that if the kernel of an endomorphism $\phi$ is not trivial then the growth rate $\mathrm{GR}(\phi)$ equals either $1$ or the spectral radius of the induced map on the abelianization. It turns out that every monomorphism $\phi$ of $\mathcal{H}_n$ determines a unique natural number $\ell$ such that $\phi(\mathcal{H}_n)$ is generated by translations with the same translation length $\ell$. We use this to show that $\mathrm{GR}(\phi)$ of a monomorphism $\phi$ of $\mathcal{H}_n$ is precisely $\ell$ for all $2\leq n$.