Vol. 298, No. 1, 2019

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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

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.

geometric group theory, translation-like actions, hyperbolic groups
Mathematical Subject Classification 2010
Primary: 20F65, 20F67
Received: 6 March 2018
Revised: 18 June 2018
Accepted: 19 June 2018
Published: 2 February 2019
David Bruce Cohen
Department of Mathematics
University of Chicago
Chicago, IL
United States