The comparison property of amenable groups

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\mathit{\text{subequivalent}}to$\mathsf B$(wewrite$\mathsf A\preccurlyeq \mathsf B$),ifthereexistsafinitepartition$\mathsf A=\bigcup_{i=1}^k \mathsf A_i$of$\mathsf A$intoclopensetsandthereareelements$g_1,g_2,\dots,g_k$in$G$suchthat$g_1(\mathsf A_1), g_2(\mathsf A_2),\dots, g_k(\mathsf A_k)$aredisjointsubsetsof$\mathsf B$. 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.