Abstract

For finitely generated groups $H$ and $G$, equipped with word metrics, a translation-like action of $H$ on $G$ is a free action such that each element of $H$ acts by a map which has finite distance from the identity map in the uniform metric. For example, if $H$ is a subgroup of $G$, then right translation by elements of $H$ yields a translation-like action of $H$ on $G$. Whyte asked whether a group having no translation-like action by a Baumslag–Solitar group must be hyperbolic, where the free abelian group of rank $2$ is understood to be a Baumslag–Solitar group. We show that the converse question has a negative answer, and in particular the fundamental group of a closed hyperbolic 3-manifold admits a translation-like action by the free abelian group of rank $2$.

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Mathematical Subject Classification 2010

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