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For finitely generated groups H and G, equipped with word metrics, atranslation-like action of H on G is a free action such that each element of Hacts by a map which has finite distance from the identity map in the uniformmetric. For example, if H is a subgroup of G, then right translation byelements of H yields a translation-like action of H on G. Whyte asked whether agroup with no translation-like action by a Baumslag-Solitar group must behyperbolic, where the free abelian group of rank 2 is understood to be aBaumslag-Solitar group. We show that the converse of this conjecture is false,and in particular the fundamental group of a closed hyperbolic 3-manifoldadmits a translation-like action by the free abelian group of rank 2.

A counterexample to the easy direction of the geometric Gersten conjecture