Rotation numbers in Thompson-Stein groups and applications

Rotation Numbers of Elements in Thompson's Group ${\bf T}$

We give a simple combinatorial proof that the rotation number for each element in Thompson's group ${\bf T}$ is rational.

将 Stein 群作为拓扑满群进行研究

Studying Stein's Groups as Topological Full Groups

本文将 Stein 对 Thompson 群 V 的推广纳入拓扑满群框架，统一证明斜率群有限生成时其单导子群有限生成，并计算多类例子的阿贝尔化与有理同调。

分类 Stein 群

Classifying Stein's groups

本文研究推广 Higman–Thompson 群的 Stein 群，借助充裕群胚、拓扑满群与 $H^1$-刚性完成秩至少 2 情形的分类，并用吸引元处理秩 1 情形。

Denominator bounds in Thompson-like groups and flows

Let T denote Thompson's group of piecewise 2-adic linear homeomorphisms of the circle. Ghys and Sergiescu showed that the rotation number of every element of T is rational, but their proof is very indirect. We give here a short, direct proof using train tracks, which generalizes to PL homeomorphism of the circle with rational break points and derivatives which are powers of some fixed integer, and also to certain flows on surfaces which we call "Thompson-like". We also obtain an explicit upper bound on the smallest period of a fixed point in terms of data which can be read off from the combina…

Ideal structure of the C*-algebra of Thompson group T

In a recent paper Uffe Haagerup and Kristian Knudsen Olesen show that for Richard Thompson's group $T$, if there exists a finite set $H$ which can be decomposed as disjoint union of sets $H_1$ and $H_2$ with $\sum_{g\in H_1}π(g)=\sum_{h\in H_2}π(h)$ and such that the closed ideal generated by $\sum_{g\in H_1}λ(g)-\sum_{h\in H_2}λ(h)$ coincides with $C^*_λ(T)$, then the Richard Thompson group $F$ is not amenable. In particular, if $C_λ^*(T)$ is simple then $F$ is not amenable. Here we prove the converse, namely, if $F$ is not amenable then we can find two sets $H_1$ and $H_2$ with the above pro…

一个在重整化下具有周期性的圆周分段线性同胚

A piecewise linear homeomorphism of the circle which is periodic under renormalization

本研究构造了一个斜率为2/3幂次、旋转数为√2-1且重整化后周期的圆周分段线性同胚，否定了Calegari的相关问题，证明F_{2/3}无法嵌入F，并给出其变体问题的肯定性证据与计算观测结果。