In this paper we study geometric versions of Burnside’s Problem and the von
Neumann Conjecture. This is done by considering the notion of a translation-like
action. Translation-like actions were introduced by Kevin Whyte as a geometric
analogue of subgroup containment. Whyte proved a geometric version of the von
Neumann Conjecture by showing that a finitely generated group is nonamenable if
and only if it admits a translation-like action by any (equivalently every) nonabelian
free group. We strengthen Whyte’s result by proving that this translation-like action
can be chosen to be transitive when the acting free group is finitely generated. We
furthermore prove that the geometric version of Burnside’s Problem holds true. That
is, every finitely generated infinite group admits a translation-like action by
. This
answers a question posed by Whyte. In pursuit of these results we discover an
interesting property of Cayley graphs: every finitely generated infinite group
has some locally finite Cayley graph having a regular spanning
tree. This regular spanning tree can be chosen to have degree
(and hence be a bi-infinite Hamiltonian path) if and only if
has
finitely many ends, and it can be chosen to have any degree greater than
if and
only if
is
nonamenable. We use this last result to then study tilings of groups. We define a
general notion of polytilings and extend the notion of MT groups and ccc groups to
the setting of polytilings. We prove that every countable group is poly-MT and every
finitely generated group is poly-ccc.
