Burnside's Problem, spanning trees, and tilings

Abstract

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 $\mathbb{Z}$. 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.

24 pages; 1 figure; minor revisions

Cited by in corpus (8)

• Translation-like Actions and Aperiodic Subshifts on Groups
• On the entropies of subshifts of finite type on countable amenable groups
• Harmonic functions with finite $p$-energy on lamplighter graphs are constant
• A remark on the connectedness of spheres in Cayley graphs
• The Burnside problem for locally compact groups
• A counterexample to the easy direction of the geometric Gersten conjecture
• Translation-like actions yield regular maps
• Dead ends on wreath products and lamplighter groups