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 $\phi$ 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 $2\leq n$.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 arXiv:
 arXiv:1512.00079
 Bibcode:
 2015arXiv151200079L
 Keywords:

 Mathematics  Group Theory;
 20E36;
 20F28;
 20K30
 EPrint:
 36 pages, 4 figures