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 emph{subequivalent} to $mathsf B$ (we write $mathsf Apreccurlyeq mathsf B$), 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 $mathsf Apreccurlyeq mathsf B$. 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.