Tilings of amenable groups

By Tomasz Downarowicz, Dawid Huczek, Guohua Zhang. Journal für die Reine und Angewandte Mathematik, 2015

103 citations recorded by the source corpus.

We prove that for any infinite countable amenable group G, any {\varepsilon>0} and any finite subset {K\subset G} , there exists a tiling (partition of G into finite “tiles” using only finitely many “shapes”), where all the tiles are {(K,\varepsilon)} -invariant. Moreover, our tiling has topological entropy zero (i.e., subexponential complexity of patterns). As an application, we construct a free action of G (in the sense that the mappings, associated to elements of G other than the unit, have no fixed points) on a zero-dimensional space, such that the topological entropy of this action is zero.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/fd1fc96605a660d4471f686eef9ea4d1e182d6d8

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In Discrete Mathematics and Combinatorics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Dimension, comparison, and almost finiteness (2017, background): “there is a ﬁnite collection {(Vi,Si)}i∈I of open towers covering Xsuch that Si is (K,δ)-invariant for every i∈Iand the family {SiVi}i∈I has chromatic number at most d+1. Proof. By the main theorem of [7] there exist nonempty (K,δ)-invariant ﬁnite sets F1,...,Fn⊆Gand sets C1,...,Cn⊆Gsuch that G= Gn k”
Almost Finiteness and the Small Boundary Property (2018, background): “9 in [10] and can be established by the same argument.”
Borel asymptotic dimension and hyperfinite equivalence relations (2020, background): “fact that every countable amenable group can be tiled by Følner sets[8].”