The comparison property of amenable groups

2017/12/14 by Tomasz Downarowicz, Downarowicz, Tomasz, Guohua Zhang +1 · 2 citations
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #Cellular Automata and Applications #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1712.05129

Abstract

Let a countable amenable group G act on a \zd compact metric space X. For two clopen subsets \mathsf A and \mathsf B of X we say that \mathsf A is subequivalent to \mathsf B (we write \mathsf A\preccurlyeq \mathsf B), if there exists a finite partition \mathsf A=\bigcup_i=1^k \mathsf A_i of \mathsf A into clopen sets and there are elements g₁,g₂,…,g_k in G such that g₁(\mathsf A₁), g₂(\mathsf A₂),…, g_k(\mathsf A_k) are disjoint subsets of \mathsf B. We say that the action admits comparison if for any clopen sets \mathsf A, \mathsf B, the condition, that for every G-invariant probability measure μ on X we have the sharp inequality μ(\mathsf A)

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• The type semigroup, comparison and almost finiteness for ample groupoids
• Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed Algorithms

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