부산대학교 도서관

The authors study free continuous actions of countable amenable groups on compact metrizable spaces and investigate connections between almost finiteness and the small boundary property. Amenability of the acting group typically manifests itself via various forms of the orbit tilings, which in the framework of this paper are expressed through the so-called {\it open castles}. Given an action $G\curvearrowright X$ on a compact metrizable space, an {\it open tower} is a pair $(V,S)$, where $V\subseteq X$ is open, $S\subseteq G$ is finite, and sets $sV$, $s\in S$, are pairwise disjoint. The set $S$ in the tower is said to be its {\it shape}. An {\it open castle} is a finite collection of open towers $\{(V_i,S_i)\}_{i\in I}$ such that all the towers $S_iV_i$, $i\in I$, are pairwise disjoint. Given a finite $K\subseteq G$ and a $\delta>0$, a set $S\subseteq G$ is said to be $(K,\delta)$-{\it invariant} if $|KS\triangle S|<\delta|S|$. \par The general idea of an almost finiteness concept is existence of open castles where the shapes $S_i$ are almost invariant under a given finite subset of the acting group and the leftover set $X\setminus\bigsqcup_{i\in I}S_iV_i$ is small. Two particular instances of this concept, which differ in the way smallness of the leftover set is defined, appear in this work. \par Let $G\curvearrowright X$ be a free continuous action of a countable amenable group on a compact metrizable space. Let $M_G(X)$ denote the space of invariant measures of the actions. Given subsets $A,B\subseteq X$ we write $ A\prec B$ to denote that for every closed $C\subseteq A$ there is an open cover $C\subseteq U_1\cup\cdots\cup U_n$ and elements $s_i\in G$, $i\le n$, such that $s_iU_i$ are pairwise disjoint and $s_iU_i\subseteq B$ for all $i\le n$. \par The action is said to be {\it almost finite} if for every $n\in\Bbb{N}$, finite $K\subseteq G$, and $\delta>0$ there are \roster \item an open castle $\{(V_i,S_i)\}_{i\in I}$ whose shapes are $(K,\delta)$-invariant and whose levels $sV_i$, $s\in S_i$, have diameter less than $\delta$, \item sets $S'_i\subseteq S_i$ such that $|S_i'|<|S_i|/n$ and $X\setminus\bigsqcup_{i\in I}S_iV_i\prec\bigsqcup_{i\in I}S_i'V_i$. \endroster In less formal terms, the second condition says that the leftover set admits many (at least~$n$) disjoint piecewise translations into the castle itself. A weaker condition of requiring the leftover to have small measure brings us to the following notion. The action $G\curvearrowright X$ is said to be {\it almost finite in measure} if for all finite $K\subseteq G$, and $\delta,\epsilon>0$, there is an open castle $\{(V_i,S_i)\}_{i\in I}$ that satisfies item (1) above and also $$ \mu\left(X\setminus\bigsqcup_{i\in I}S_iV_i\right)\ge 1-\epsilon\quad\text{for all}\ \mu\in M_G(X). $$ \par Based on the ergodic theoretical ideas developed in [D.~S. Ornstein and B. Weiss, J. Analyse Math. {\bf 48} (1987), 1--141; MR0910005], the authors show (Theorem 3.13) that all free actions of amenable groups on {\it zero-dimensional} compact metrizable spaces are almost finite in measure. When it comes to actions on general compact metrizable spaces, almost finiteness in measure turns out to be equivalent (Theorem 5.6) to the {\it small boundary property}, i.e., existence of a basis for the topology on $X$ consisting of open sets $U$ with $\mu(\partial U)=0$ for all $\mu\in M_G(X)$. \par Let us say that the action $G\curvearrowright X$ has {\it comparison} if $A\prec B$ holds for all open ${A,B\subseteq X}$ such that $\mu(A)<\mu(B)$ for all $\mu\in M_G(X)$. In Theorem 6.1 it is shown that the action $G\curvearrowright X$ has the small boundary property and comparison if and only if it is almost finite. \par Two results are given that guarantee almost finiteness of the action. First, it is proved in Corollary 7.7 that if all actions of an amenable group $G$ on {\it zero-dimensional} compact metrizable spaces are almost finite, then so are all of its actions on {\it finite@-dimensional} compact metrizable spaces. Second, Theorem 8.1 concludes that if all finitely generated subgroups of $G$ have subexponential growth then all free actions of $G$ on finite@-dimensional spaces are almost finite. \par The notion of almost finiteness is motivated by the theory of $C^*$-algebras, and the paper puts the aforementioned results into an operator algebraic perspective. For example, the authors show in Theorem 8.2 that the crossed products that arise from free minimal actions of groups of locally subexponential growth on finite-dimensional spaces are classified by the Elliott invariant and are simple ASH (approximately subhomogeneous) algebras of topological dimension at most 2. The paper is concluded with a discussion of the connections with the Toms-Winter Conjecture, where the authors show (Theorem 9.4) that crossed products $C(X)\rtimes G$ have uniform property $\Gamma$ whenever the free action of an infinite amenable group $G\curvearrowright X$ has the small boundary property.