可数群作用的比较性与几乎有限性
Comparison and Almost Finiteness for Actions of Amenable Groups
浏览论文内容
中文总结 AI 辅助
本文证明可数无限离散顺从群在零维或可度量空间上的作用具有动力学比较性，并由此建立几乎有限性，解决多个猜想与开放问题。
中文摘要 AI 辅助
我们证明，可数无限离散顺从群在非空紧致豪斯多夫零维空间上的每一个作用都具有动力学比较性，无需假设极小性、自由性或可度量化。更精确地说，在所有不变概率测度下的严格不等式蕴含了带序单位余项的闭开型半群中的比较性。对于极小康托作用，闭开型半群是可消的且几乎无穿孔的，其到余不变群正锥的典范映射是序幺半群的同构。这些结果回答了梅勒雷的可消性问题及其比较性问题的顺从情形。对于极小康托作用，我们还证明了相伴仿射求值映射的良性，回答了第一作者的一个问题，并得到了同一康托空间上具有完全相同不变概率测度的极小同胚。然后我们确定了仅从仿射求值恢复闭开等可分解性的精确障碍。该障碍由无穷小子群给出：对于每个非空真闭开集 $A$，它参数化了与 $A$ 具有相同仿射求值的闭开集的拓扑全群轨道。将比较性与克尔和绍博的定理相结合，我们证明了可数无限离散顺从群在具有拓扑小边界性质的非空紧致可度量空间上的每一个自由作用都是几乎有限的。特别地，有限维紧致可度量空间上的每一个自由作用都是几乎有限的，解决了纳雷什金提出的几乎有限性猜想。若该作用也是极小的，则其约化交叉积是 $\mathcal Z$-稳定的，具有至多一的核维数，并满足汤姆斯-温特猜想中的所有正则性条件。
英文摘要
We prove that every action of a countably infinite discrete amenable group on a nonempty compact Hausdorff zero-dimensional space has dynamical comparison, without assuming minimality, freeness, or metrizability. More precisely, strict inequalities under all invariant probability measures imply comparison in the clopen type semigroup with an order-unit remainder. For minimal Cantor actions, the clopen type semigroup is cancellative and almost unperforated, and its canonical map onto the positive cone of the coinvariant group is an isomorphism of ordered monoids. These results answer Melleray's cancellation question and the amenable case of his comparison question. For minimal Cantor actions, we also prove goodness of the associated affine evaluation map and obtain a minimal homeomorphism of the same Cantor space with the same invariant probability measures. We identify the precise obstruction to recovering clopen equidecomposability from affine evaluation alone. This obstruction is given by the infinitesimal subgroup: for every nonempty proper clopen set $A$, it parametrizes the topological-full-group orbits of clopen sets having the same affine evaluation as $A$. Combining comparison with the theorems of Kerr and Szabó, we prove that every free action of a countably infinite discrete amenable group on a nonempty compact metrizable space with the topological small boundary property is almost finite. This settles Naryshkin's almost-finiteness conjecture for finite-dimensional compact metrizable spaces. If the action is also minimal, its reduced crossed product is $\mathcal Z$-stable, has nuclear dimension at most one, and satisfies all the regularity conditions in the Toms--Winter conjecture.
发表机构
- Tel Aviv University（特拉维夫大学）
- Dalian University of Technology（大连理工大学）
- Institute of Mathematics, Polish Academy of Sciences（波兰科学院数学研究所）
机构由 AI 辅助整理，请以论文原文为准。
