The group of almost-periodic homeomorphisms of the real line

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。

关于流形同胚群的Burnside问题的注记

A note on the Burnside problem for homeomorphism groups of manifolds

本文研究紧连通流形同胚群的Burnside问题，证明表面非例外时同胚群恒等分量torsion-free，扩展映射类群Tits猜想，恢复并扩展Guelman和Liousse定理至非定向表面，证明圆同胚群有限生成周期子群为有限cyclic，并讨论有边界的流形及双柄体的开放问题。

Jordan property for homeomorphism groups and almost fixed point property

We study properties of continuous finite group actions on topological manifolds that hold true, for any finite group action, after possibly passing to a subgroup of index bounded above by a constant depending only on the manifold. These include the Jordan property, the almost fixed point property, as well as bounds on the discrete symmetry group. Most of our results apply to manifolds satisfying some restriction such as having nonzero Euler characteristic or having the integral homology of a sphere. For an arbitrary topological manifold $X$ such that $H_*(X;{\mathbf Z})$ is finitely generated,…

Braid groups are almost co-Hopfian

Let B_n be the braid group on n > 3 strands. We prove that B_n modulo its center is co-Hopfian. We then show that any injective endomorphism of B_n is geometric in the sense that it is induced by a homeomorphism of a punctured disk. We further prove that any injection from B_n to B_n+1 is geometric. Additionally, we obtain analogous results for mapping class groups of punctured spheres. The methods use Thurston's theory of surface homeomorphisms and build upon work of Ivanov and McCarthy.

穿孔圆盘上的强Nielsen等价性

Strong Nielsen equivalence on the punctured disc

本文研究穿孔圆盘上保持定向同胚的周期点强Nielsen等价性，在辫子理论中给出两个周期点强Nielsen等价的充要条件，并应用于Jiang--Zheng迹公式，证明强制周期轨道分属不同强Nielsen类。

Partial periodic quotient of groups acting on a hyperbolic space

In this article, we construct partial periodic quotients of groups which have a non-elementary acylindrical action on a hyperbolic space. In particular, we provide infinite quotients of mapping class groups where a fixed power of every pseudo-Anosov homeomorphism is identified with a periodic or reducible element.

Algorithms and Classification in Groups of Piecewise-Linear Homeomorphisms

This manuscript represents the author's PhD dissertation thesis.The first part studies decision problems in Thompson's groups F,T,V and some generalizations. The simultaneous conjugacy problem is determined to be solvable for Thompson's group F and suitable larger groups of piecewise-linear homeomorphisms of the unit interval. We determine algorithms to compute roots and centralizers in these groups and to detect periodic points and their behavior by looking at a particular diagram associated to an element. In the second part, we describe the structure of subgroups of the group of all homeomor…

3-流形中的投射有限几乎刚性

Profinite almost rigidity in 3-manifolds

该研究证明带空或环面边界的紧致可定向3-流形具投射有限几乎刚性，结合外围结构可唯一确定混合3-流形Seifert部分同胚类型，否则无法唯一确定，证明基于JSJ分解。

On the stabilisers of points in groups with micro-supported actions

Given a group $G$ of homeomorphism of a first-countable Hausdorff space $\mathcal{X}$, we prove that if the action of $G$ on $\mathcal{X}$ is minimal and has rigid stabilisers that act locally minimally, then the neighbourhood stabilisers of any two points in $\mathcal{X}$ are conjugated by a homeomorphism of $\mathcal{X}$. This allows us to study stabilisers of points in many classes of groups, such as topological full groups of Cantor minimal systems, Thompson groups, branch groups, and groups acting on trees with almost prescribed local actions.

Which infinite abelian groups admit an almost maximally almost-periodic group topology?

A topological group G is said to be almost maximally almost-periodic if its von Neumann radical is non-trivial, but finite. In this paper, we prove that every abelian group with an infinite torsion subgroup admits a (Hausdorff) almost maximally almost-periodic group topology. Some open problems are also formulated.