부산대학교 도서관

The authors study free continuous actions of countable amenable groupson compact metrizable spaces and investigate connections between almostfiniteness and the small boundary property. Amenability of the actinggroup typically manifests itself via various forms of the orbittilings, which in the framework of this paper are expressed through theso-called {\it open castles}. Given an action $G\curvearrowright X$ ona 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 beits {\it shape}. An {\it open castle} is a finite collection of opentowers $\{(V_i,S_i)\}_{i\in I}$ 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 opencastles where the shapes $S_i$ are almost invariant under a givenfinite subset of the acting group and the leftover set$X\setminus\bigsqcup_{i\in I}S_iV_i$ is small. Two particular instancesof this concept, which differ in the way smallness of the leftover setis defined, appear in this work.\par Let $G\curvearrowright X$ be a free continuous action of a countableamenable group on a compact metrizable space. Let $M_G(X)$ denote thespace of invariant measures of the actions. Given subsets $A,B\subseteqX$ 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 $sV_i$, $s\in S_i$, havediameter 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 setadmits many (at least~$n$) disjoint piecewise translations into thecastle itself. A weaker condition of requiring the leftover to havesmall measure brings us to the following notion. The action$G\curvearrowright X$ is said to be {\it almost finite in measure} iffor all finite $K\subseteq G$, and $\delta,\epsilon>0$, there is an opencastle $\{(V_i,S_i)\}_{i\in I}$ that satisfies item (1) above and also$$\mu\left(X\setminus\bigsqcup_{i\in I}S_iV_i\right)\ge1-\epsilon\quad\text{for all}\ \mu\in M_G(X).$$\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 freeactions of amenable groups on {\it zero-dimensional} compact metrizablespaces are almost finite in measure. When it comes to actions ongeneral compact metrizable spaces, almost finiteness in measure turnsout to be equivalent (Theorem 5.6) to the {\it small boundaryproperty}, i.e., existence of a basis for the topology on $X$consisting of open sets $U$ with $\mu(\partial U)=0$ for all $\mu\inM_G(X)$.\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 shownthat the action $G\curvearrowright X$ has the small boundary propertyand 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 amenablegroup $G$ on {\it zero-dimensional} compact metrizable spaces arealmost finite, then so are all of its actions on {\it finite@-dimensional} compact metrizable spaces. Second, Theorem 8.1concludes that if all finitely generated subgroups of $G$ havesubexponential growth then all free actions of $G$ onfinite@-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 anoperator algebraic perspective. For example, the authors show inTheorem 8.2 that the crossed products that arise from free minimalactions of groups of locally subexponential growth onfinite-dimensional spaces are classified by the Elliott invariant andare simple ASH (approximately subhomogeneous) algebras of topologicaldimension at most 2. The paper is concluded with a discussion of theconnections with the Toms-Winter Conjecture, where the authors show(Theorem 9.4) that crossed products $C(X)\rtimes G$ have uniformproperty $\Gamma$ whenever the free action of an infinite amenablegroup $G\curvearrowright X$ has the small boundary property.