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Growth rate of endomorphisms of Houghton's groups by Lee, Jong Bum; Lee, Sang Rae is a scholarly article available to read on EtoBox.

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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 $2\leq n$.

Author

Lee, Jong Bum; Lee, Sang Rae

Published

2015

Language

EN