For finitely generated groups
and
,
equipped with word metrics, a translation-like action of
on
is a free action such
that each element of
acts by a map which has finite distance from the identity map in the uniform metric. For example,
if
is a subgroup of
, then right translation
by elements of
yields a
translation-like action of
on
.
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
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
.
Keywords
geometric group theory, translation-like actions,
hyperbolic groups
