Almost finiteness and homology of certain non-free actions

k-同调有限性与双曲群

K-homological finiteness and hyperbolic groups

受有关闭流形的经典事实的启发，我们在K-同调中引入了一个强有限性。我们称一个C* 代数具有一致可和的K-同调，如果它的所有K-同调类都可以用在同一稠密子代数上可和的Fredholm模来表示，并且具有相同的可和度.证明了与双曲群相关的两类C* 代数-代数--边界作用的C*-交叉积和约化群C* 代数--具有一致可和的K-同调.我们提供了明确的可和度，以及明确的K-同调类的可和代表。

Almost finiteness for general etale groupoids and its applications to stable rank of crossed products

We extend Matui's notion of almost finiteness to general etale groupoids and show that the reduced groupoid C*-algebras of minimal almost finite groupoids have stable rank one. The proof follows a new strategy, which can be regarded as a local version of the large subalgebra argument. The following three are the main consequences of our result. (i) For any group of (local) subexponential growth and for any its minimal action admitting a totally disconnected free factor, the crossed product has stable rank one. (ii) Any countable amenable group admits a minimal action on the Cantor set all whos…

Almost finiteness, comparison, and tracial $\mathcal{Z}$-stability

Inspired by Kerr's work on topological dynamics, we define tracial $\mathcal{Z}$-stability for sub-$C^*$-algebras. We prove that for a countable discrete amenable group $G$ acting freely and minimally on a compact metrizable space $X$, tracial $\mathcal{Z}$-stability for the sub-$C^*$-algebra $(C(X)\subseteq C(X)\rtimes G)$ implies that the action has dynamical comparison. Consequently, tracial $\mathcal{Z}$-stability is equivalent to almost finiteness of the action, provided that the action has the small boundary property.

Dimension, comparison, and almost finiteness

We develop a dynamical version of some of the theory surrounding the Toms-Winter conjecture for simple separable nuclear C*-algebras and study its connections to the C*-algebra side via the crossed product. We introduce an analogue of hyperfiniteness for free actions of amenable groups on compact spaces and show that it plays the role of Z-stability in the Toms-Winter conjecture in its relation to dynamical comparison, and also that it implies Z-stability of the crossed product. This property, which we call almost finiteness, generalizes Matui's notion of the same name from the zero-dimensiona…

初等可 amenability 与近有限性

Elementary amenability and almost finiteness

本文证明了可数无限初等可 amenable 群在有限维紧致可度量空间上的自由连续作用是近有限的，从而得出交叉积是 $\mathcal{Z}$-稳定的，并通过 Elliott 不变量分类。

The type semigroup, comparison and almost finiteness for ample groupoids

We prove that a minimal second countable ample groupoid has dynamical comparison if and only if its type semigroup is almost unperforated. Moreover, we investigate to what extent a not necessarily minimal almost finite groupoid has an almost unperforated type semigroup. Finally, we build a bridge between coarse geometry and topological dynamics by characterizing almost finiteness of the coarse groupoid in terms of a new coarsely invariant property for metric spaces, which might be of independent interest in coarse geometry. As a consequence, we are able to construct new examples of almost fini…