## Geometry & Topology

- Geom. Topol.
- Volume 18, Number 1 (2014), 179-210.

### 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 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 $\mathbb{Z}$. 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 locally finite 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 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.

#### Article information

**Source**

Geom. Topol., Volume 18, Number 1 (2014), 179-210.

**Dates**

Received: 6 April 2011

Revised: 4 November 2011

Accepted: 25 August 2013

First available in Project Euclid: 20 December 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.gt/1513732722

**Digital Object Identifier**

doi:10.2140/gt.2014.18.179

**Mathematical Reviews number (MathSciNet)**

MR3158775

**Zentralblatt MATH identifier**

1338.20041

**Subjects**

Primary: 20F65: Geometric group theory [See also 05C25, 20E08, 57Mxx]

Secondary: 05C25: Graphs and abstract algebra (groups, rings, fields, etc.) [See also 20F65] 05C63: Infinite graphs

**Keywords**

Burnside's Problem von Neumann conjecture Hamiltonian path Lovász conjecture spanning tree tile tiling MT ccc translation-like action

#### Citation

Seward, Brandon. Burnside's Problem, spanning trees and tilings. Geom. Topol. 18 (2014), no. 1, 179--210. doi:10.2140/gt.2014.18.179. https://projecteuclid.org/euclid.gt/1513732722