关于拓扑全群的顺从性与LEF性质
On Amenability and LEF-ness for topological full groups
AI总结：
本文证明了拓扑自由远端作用的拓扑全群顺从性等价于底群顺从，并推广了LEF性质到剩余有限作用，进而证明自由群上某些Toeplitz子位移的拓扑全群是LEF且sofic。
摘要：
本文证明了对于可数离散群$G$在康托集上的拓扑自由远端作用$\alpha$，其拓扑全群$[[\alpha]]$是顺从的当且仅当$G$是顺从的。这推广了Cortez和Medynets的一个结果。此外，通过计算Elek和Monod提供的某些例子的拓扑熵，我们获得了关于零拓扑熵作用的拓扑全群顺从性的否定结果。进一步地，我们证明了康托集上极小拓扑自由剩余有限作用$\alpha$的拓扑全群$[[\alpha]]$可局部嵌入有限群类（LEF）。这推广了Grigorchuk和Medynets在极小$\Z$-作用情形下先前获得的一个结果。作为应用，我们证明了自由群上某些Toeplitz子位移的拓扑全群是LEF的，因此也是sofic的。
英文摘要：
In this paper, we prove that for a topologically free 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 generalizes a result by Cortez and Medynets. In addition, 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 extends a result previously obtained by Grigorchuk and Medynets in the case of minimal $\Z$-actions. As an application, we show that topological full groups of certain Toeplitz subshifts on free groups are LEF and therefore sofic.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
