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 F{\o}lner properties, which are factors of the action. Also, the theory of symbolic extensions, known for $\mathbb z$-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 $\mathbb z$ and $\mathbb z^d$ 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.