丰富群胚、拓扑全群、代数K-理论谱与无穷环空间

Matui's AH conjecture for Graph Groupoids

We prove that Matui's AH conjecture holds for graph groupoids of infinite graphs. This is a conjecture which relates the topological full group of an ample groupoid with the homology of the groupoid. Our main result complements Matui's result in the finite case, which makes the AH conjecture true for all graph groupoids covered by the assumptions of said conjecture. Furthermore, we observe that for arbitrary graphs, the homology of a graph groupoid coincides with the $K$-theory of its groupoid $C^*$-algebra.

拓扑全群的稳定子与极大子群

Stabilizers and maximal subgroups of topological full groups

该研究将Grigorchuk等人的极大稳定子定理推广到含换位子群的子群，证明了极小作用稳定子的空间刚性，实现了AF群胚的限制，得到连续多个不同构极大稳定子及C*-代数的AF子商。

$C^*$-simplicity and representations of topological full groups of groupoids

Given an ample groupoid $G$ with compact unit space, we study the canonical representation of the topological full group $[[G]]$ in the full groupoid $C^*$-algebra $C^*(G)$. In particular, we show that the image of this representation generates $C^*(G)$ if and only if $C^*(G)$ admits no tracial state. The techniques that we use include the notion of groups covering groupoids. As an application, we provide sufficient conditions for $C^*$-simplicity of certain topological full groups, including those associated with topologically free and minimal actions of non-amenable and countable groups on t…

Algebraic properties of G-groupoids

The main purpose of this paper is to give a new definition for the notion of group-groupoid. Also, several basic properties of group-groupoids are established.

Homology of odometers

We compute the homology groups of transformation groupoids associated with odometers and show that certain infinite dihedral group odometers give rise to counterexamples to the HK conjecture, which relates the homology of an essentially principal, minimal, ample groupoid $G$ with the K-theory of $C^*_r(G)$. We also show that transformation grupoids of odometers satisfy the AH conjecture.

群与群胚自相似作用的同调与K-理论

Homology and K-theory for self-similar actions of groups and groupoids

本文为自相似群作用构造长正合序列，计算其群胚同调和C*-代数K-理论，并证明包含Grigorchuk群的Röver单群有理无环但Schur乘子非平凡。

Steinberg代数和C*-代数中拓扑全群的表示

Representing topological full groups in Steinberg algebras and C*-algebras

本文研究了ample Hausdorff群胚的拓扑全群在Steinberg代数及C*-代数中的自然表示，刻画了其单射性，并揭示了离散群胚下该表示在满和约化C*-代数中的稠密性差异。

Steinberg代数的同调

Homology of Steinberg algebras

本文研究了ample群胚Steinberg代数的同调不变量，计算了特定群胚的Hochschild与循环同调，并利用Dennis迹建立了K理论到群胚同调的映射，最后针对Exel-Pardo群胚给出了具体计算。

Topological full groups of etale groupoids

This is a survey of the recent development of the study of topological full groups of etale groupoids on the Cantor set. Etale groupoids arise from dynamical systems, e.g. actions of countable discrete groups, equivalence relations. Minimal Z-actions, minimal Z^N-actions and one-sided shifts of finite type are basic examples. We are interested in algebraic, geometric and analytic properties of topological full groups. More concretely, we discuss simplicity of commutator subgroups, abelianization, finite generation, cohomological finiteness properties, amenability, the Haagerup property, and so…

Short and biased introduction to groupoids

The algebraic part of approach to groupoids started by S. Zakrzewski is presented.

关于群概念的三种推广：groupoid、广义群和几乎groupoid

On three generalizations of the group concept: groupoid, generalized group and almost groupoid

本文介绍了群概念的三种推广（groupoid、广义群和几乎groupoid）的定义、主要性质、构造及相关例子。

Vector Groupoids

The main purpose of this paper is to study the vector groupoids. This is an algebraic structure which combines the concepts of Brandt groupoid and vector space such that these are compatible.

Ampleness in the free group

We show that the theory of the free group -- and more generally the theory of any torsion-free hyperbolic group -- is $n$-ample for any $n\geq 1$. We give also an explicit description of the imaginary algebraic closure in free groups.

Delta-groupoids and ideal triangulations

A Delta-groupoid is an algebraic structure which axiomatizes the combinatorics of a truncated tetrahedron. By considering two simplest examples coming from knot theory, we illustrate how can one associate a Delta-groupoid to an ideal triangulation of a three-manifold. We also describe in detail the rings associated with the Delta-groupoids of these examples.