A counterexample to the easy direction of the geometric Gersten conjecture

TL;DR

This paper provides a counterexample to the conjecture that groups without translation-like actions by Baumslag-Solitar groups must be hyperbolic, showing that some hyperbolic 3-manifold groups admit such actions.

Contribution

It demonstrates that the converse of Whyte's conjecture is false by constructing a counterexample involving hyperbolic 3-manifold groups.

Findings

The fundamental group of a closed hyperbolic 3-manifold admits a translation-like action by Z^2.

Counterexample disproves the conjecture that absence of certain actions implies hyperbolicity.

Shows that hyperbolic groups can admit translation-like actions by non-hyperbolic groups.

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 with 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 of this conjecture is false, 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.