#### Vol. 298, No. 1, 2019

 Recent Issues Vol. 298: 1  2 Vol. 297: 1  2 Vol. 296: 1  2 Vol. 295: 1  2 Vol. 294: 1  2 Vol. 293: 1  2 Vol. 292: 1  2 Vol. 291: 1  2 Online Archive Volume: Issue:
 The Journal Subscriptions Editorial Board Officers Special Issues Submission Guidelines Submission Form Contacts Author Index To Appear ISSN: 1945-5844 (e-only) ISSN: 0030-8730 (print) Other MSP Journals
A counterexample to the easy direction of the geometric Gersten conjecture.

### David Bruce Cohen

Vol. 298 (2019), No. 1, 27–31
DOI: 10.2140/pjm.2019.298.27
##### 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 $2$.

##### Keywords
geometric group theory, translation-like actions, hyperbolic groups
##### Mathematical Subject Classification 2010
Primary: 20F65, 20F67