  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 non-amenable if and only if it admits a translation-like action by any
(equivalently every) non-abelian 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 $\Z$. 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
$G$ has some Cayley graph having a regular spanning tree. This regular spanning
tree can be chosen to have degree 2 (and hence be a bi-infinite Hamiltonian
path) if and only if $G$ has finitely many ends, and it can be chosen to have
any degree greater than 2 if and only if $G$ is non-amenable. 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.
