# zbMATH — the first resource for mathematics

Burnside’s problem, spanning trees and tilings. (English) Zbl 1338.20041
Summary: 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.

##### MSC:
 20F65 Geometric group theory 05C25 Graphs and abstract algebra (groups, rings, fields, etc.) 05C63 Infinite graphs 20F05 Generators, relations, and presentations of groups 20F50 Periodic groups; locally finite groups 43A07 Means on groups, semigroups, etc.; amenable groups
Full Text:
##### References:
 [1] I Benjamini, O Schramm, Every graph with a positive Cheeger constant contains a tree with a positive Cheeger constant, Geom. Funct. Anal. 7 (1997) 403 · Zbl 0882.05052 · doi:10.1007/PL00001625 [2] B Bollobás, Modern graph theory, Graduate Texts in Mathematics 184, Springer (1998) · Zbl 0902.05016 · doi:10.1007/978-1-4612-0619-4 [3] N Brady, Branched coverings of cubical complexes and subgroups of hyperbolic groups, J. London Math. Soc. 60 (1999) 461 · Zbl 0940.20048 · doi:10.1112/S0024610799007644 [4] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundl. Math. Wissen. 319, Springer (1999) · Zbl 0988.53001 [5] C Chou, Elementary amenable groups, Illinois Journal Math. 24 (1980) 396 · Zbl 0439.20017 · euclid:ijm/1256047608 [6] S Gao, S Jackson, B Seward, Group colorings and Bernoulli subflows · Zbl 1375.37036 · arxiv:1201.0513 [7] E S Golod, I R \vSafarevi\vc, On the class field tower, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964) 261 · Zbl 0136.02602 · mi.mathnet.ru [8] P de la Harpe, Topics in geometric group theory, Chicago Lectures in Mathematics, University of Chicago Press (2000) · Zbl 0965.20025 [9] A J Ol’, On the question of the existence of an invariant mean on a group, Uspekhi Mat. Nauk 35 (1980) 199 · Zbl 0452.20032 [10] I Pak, R Radoi, Hamiltonian paths in Cayley graphs, Discrete Math. 309 (2009) 5501 · Zbl 1229.05184 · doi:10.1016/j.disc.2009.02.018 [11] P Papasoglu, Homogeneous trees are bi-Lipschitz equivalent, Geom. Dedicata 54 (1995) 301 · Zbl 0836.05018 · doi:10.1007/BF01265344 [12] J P Serre, Trees, Springer Monographs in Mathematics, Springer-Verlag (2003) [13] B Weiss, Monotileable amenable groups (editors V Turaev, A Vershik), Amer. Math. Soc. Transl. Ser. 2 202, Amer. Math. Soc. (2001) 257 · Zbl 0982.22004 [14] K Whyte, Amenability, bi-Lipschitz equivalence, and the von Neumann conjecture, Duke Math. J. 99 (1999) 93 · Zbl 1017.54017 · doi:10.1215/S0012-7094-99-09904-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.