关于单位区间同胚离散群顺从性的Shavgulidze证明

The homeomorphism group of the first uncountable ordinal

We show that the topology of pointwise convergence on scattered spaces is compatible with the group structure of their homeomorphism group. We then establish a few topological properties of the homeomorphism group of the first uncountable ordinal, such as amenability and Roelcke-precompactness.

Subgroup dynamics and $C^\ast$-simplicity of groups of homeomorphisms

We study the uniformly recurrent subgroups of groups acting by homeomorphisms on a topological space. We prove a general result relating uniformly recurrent subgroups to rigid stabilizers of the action, and deduce a $C^*$-simplicity criterion based on the non-amenability of rigid stabilizers. As an application, we show that Thompson's group $V$ is $C^\ast$-simple, as well as groups of piecewise projective homeomorphisms of the real line. This provides examples of finitely presented $C^\ast$-simple groups without free subgroups. We prove that a branch group is either amenable or $C^\ast$-simple…

Furstenberg-Weiss型的几乎1-1扩张与顺从性检验

Almost 1-1 extensions of Furstenberg-Weiss type and test for amenability

该研究针对无限剩余有限群，证明极小等度连续Cantor系统的极小扩张存在诱导不变测度空间仿射双射的极小几乎1-1扩张，并据此得出Toeplitz子移位族可检验剩余有限群的顺从性。

关于直线上分段投射群的若干评论

Some comments on piecewise-projective groups of the line

本研究聚焦直线分段投射同胚群的顺从性问题，用显式不动点论证替代Carriere--Ghys定理，强化了其非顺从性结论，并发现群间存在不可数多个“顺从性间隙”。

理查德·汤普森F组不服从

Richard Thompson group F is not amenable

理查德·汤普森的群F是单位区间的分段线性同胚群，它有有限个断点，都是二进有理数（它们的代数是2的幂），斜率是2的幂。一个离散群G是顺从的，如果存在一个有限可加的概率测度，它在G的所有子集上定义，并且在左平移下不变. F的顺从性问题是一个著名的开放问题。本文证明了群F是不顺从的。

一些不作用于圆的算术群

Some arithmetic groups that do not act on the circle

群SL（3，Z）不能（忠实地）作用于圆（通过同胚）。我们将看到，许多其他算术群也不能作用于圆。讨论将涉及群论中的几个重要课题，如序群，顺从性，有界生成，有界上同调。第一堂课介绍了这个主题，并使用左序群的理论来证明SL（3，Z）不作用于圆。第2课讨论有界生成，证明了SL（2，Z[a]）形式的群不作用在实数直线上。第三讲和第四讲分别是顺从群和有界上同调的简要介绍。他们还解释了如何用这些思想来证明圆上的作用具有有限轨道。附录提供了所有练习的提示或参考。这些笔记是从2012年7月在帕克城数学研究所研究生暑期学校的演讲中稍微扩展的。

Thompson's group $F$ is not Liouville

We prove that random walks on Thompson's group $F$ driven by strictly non-degenerate finitely supported probability measures $μ$ have a non-trivial Poisson boundary. The proof consists in an explicit construction of two different non-trivial $μ$-boundaries. Both of them are defined in terms of the Schreier graph $Γ$ on the dyadic-rational orbit of the canonical action of $F$ on the unit interval (actually, we consider a natural embedding of $F$ into the group $PLF({\mathbb R})$ of piecewise linear homeomorphisms of the real line, and realize $Γ$ on the dyadic-rational orbit in ${\mathbb R}$). …