论文标题
拓扑完整群体的soficity，修理性和LEF-
Soficity, Amenability, and LEF-ness for topological full groups
论文作者
论文摘要
在本文中，我们研究了在cantor套件上拓扑完整的小组动作组的几个有限近似特性，使得自由点非常密集。首先，我们确定在Cantor Set上的可数离散组$ G $的远端动作$α$中，拓扑完整组$ [[α]] $在且仅当$ g $是可正常的情况下。该结果是通过一种新的方法获得的，该方法检测到有限生成的$ [[α] $的有限生成的亚组的某些SOFIC近似图序列。我们还提供了相关Følner功能的估计。接下来，我们通过计算Elek和Monod提供的某些示例的拓扑熵，从而获得拓扑完整组对具有零拓扑熵作用的拓扑结构的负面结果。此外，我们证明了最低拓扑上免费的残留量$α$的拓扑整体$ [[α] $在cantor set上是可以在有限组（LEF）等类中本地嵌入的。这概括了Grigorchuk和Medynets先前获得的结果，在最小$ \ Mathbb {Z} $ - Actions的情况下。作为一种应用，我们表明，自由组的某些Toeplitz子缩影的拓扑完整组是Lef，因此是Sofic。
In this paper, we study several finite approximation properties of topological full groups of group actions on the Cantor set such that free points are dense. Firstly, we establish that for such a distal action $α$ of a countable discrete group $G$ on the Cantor set, the topological full group $[[α]]$ is amenable if and only if $G$ is amenable. This result is obtained through a novel method that detects hyperfiniteness in certain sofic approximation graph sequences of finitely generated subgroups of $[[α]]$. We also provide estimates for related Følner functions. Next, we obtain negative results on the amenability of topological full groups for actions with zero topological entropy by calculating the topological entropy of certain examples provided by Elek and Monod. Furthermore, we demonstrate that the topological full group $[[α]]$ of a minimal topologically free residually finite action $α$ on the Cantor set is locally embeddable in the class of finite groups (LEF). This generalizes a result previously obtained by Grigorchuk and Medynets in the case of minimal $\mathbb{Z}$-actions. As an application, we show that topological full groups of certain Toeplitz subshifts on free groups are LEF and therefore sofic.
