The comparison property of amenable groups

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 \emph{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 $g_1,g_2,\dots ,g_k$ in $G$ such that $g_1({\mathsf{A}}_1),g_2({\mathsf{A}}_2),\dots ,g_k({\mathsf{A}}_k)$ are disjoint subsets of $\mathsf{B}$. We say that the action \emph{admits comparison} if for any clopen sets $\mathsf{A},\mathsf{B}$, the condition, that for every $G$-invariant probability measure $\mu$ on $X$ we have the sharp inequality $\mu (\mathsf{A})<\mu (\mathsf{B})$, implies $\mathsf{A}\preccurlyeq \mathsf{B}$. Comparison has many desired consequences for the action, such as the existence of tilings with arbitrarily good Følner properties, which are factors of the action. Also, the theory of symbolic extensions, known for $z$-actions, extends to actions which admit comparison. We also study a purely group-theoretic notion of comparison: if every action of $G$ on any zero-dimensional compact metric space admits comparison then we say that $G$ has the \emph{comparison property}. Classical groups $z$ and $z^d$ enjoy the comparison property, but in the general case the problem remains open. In this paper we prove this property for groups whose every finitely generated subgroup has subexponential growth.

overlapped by arXiv:1901.01457, and so will not be published separately (note that we have not changed the file)

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• Symbolic dynamics on amenable groups: the entropy of generic shifts
• Minimal subshifts of arbitrary mean topological dimension
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• Minimal models for actions of amenable groups
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Cited by in corpus (8)

• Nuclear dimension of simple C*-algebras
• Almost finiteness for general etale groupoids and its applications to stable rank of crossed products
• Qualitative graph limit theory. Cantor Dynamical Systems and Constant-Time Distributed Algorithms
• The type semigroup, comparison and almost finiteness for ample groupoids
• A generalized type semigroup and dynamical comparison
• An embedding theorem for subshifts over amenable groups with the comparison property
• Elementary amenability and almost finiteness
• Comparison and Simplicity of Commutator Subgroups of Full Groups