Growth rate of endomorphisms of Houghton's groups

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

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