Growth rate of endomorphisms of Houghton's groups

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 $\varphi$ is not trivial then the growth rate $\mathrm{GR}(\varphi )$ equals either $1$ or the spectral radius of the induced map on the abelianization. It turns out that every monomorphism $\varphi$ of ${\mathcal{H}}_n$ determines a unique natural number $\ell$ such that $\varphi ({\mathcal{H}}_n)$ is generated by translations with the same translation length $\ell$. We use this to show that $\mathrm{GR}(\varphi )$ of a monomorphism $\varphi$ of ${\mathcal{H}}_n$ is precisely $\ell$ for all $2\le n$.

36 pages, 4 figures

References in corpus (3)

• On the isolated points in the space of groups
• Geometry of Houghton's Groups
• Growth rate for endomorphisms of finitely generated nilpotent groups and solvable groups