Decomposing locally compact groups into simple pieces

康托尔集同胚的离散局部有限全群

Discrete locally finite full groups of Cantor set homeomorphisms

本文研究康托尔集自同胚群的紧开拓扑下局部紧分段全群的分类问题，证明其必为均匀离散群，并通过 Bratteli 图与 $K_0$ 群确定该类群的结构。

Automorphism groups of right-angled buildings: simplicity and local splittings

We show that the group of type-preserving automorphisms of any irreducible semi-regular thick right-angled building is abstractly simple. When the building is locally finite, this gives a large family of compactly generated (abstractly) simple locally compact groups. Specializing to appropriate cases, we obtain examples of such simple groups that are locally indecomposable, but have locally normal subgroups decomposing non-trivially as direct products.

康托空间同胚的局部紧分片全群

Locally compact piecewise full groups of homeomorphisms

本文研究康托空间同胚的分片全群何时具有非离散全不连通局部紧拓扑，给出交替全群紧生成的判据，并由此构造一大类导群为非离散、紧生成、开且单的 t.d.l.c. 群，统一并推广了 Neretin 等人的构造。

Piecewise strongly proximal actions, free boundaries and the Neretin groups

A closed subgroup $H$ of a locally compact group $G$ is confined if the closure of the conjugacy class of $H$ in the Chabauty space of $G$ does not contain the trivial subgroup. We establish a dynamical criterion on the action of a totally disconnected locally compact group $G$ on a compact space $X$ ensuring that no relatively amenable subgroup of $G$ can be confined. This property is equivalent to the fact that the action of $G$ on its Furstenberg boundary is free. Our criterion applies to the Neretin groups. We deduce that each Neretin group has two inequivalent irreducible unitary represen…

Finiteness properties of some groups of piecewise projective homeomorphisms

The Lodha-Moore group $G$ first arose as a finitely presented counterexample to von Neumann's conjecture. The group $G$ acts on the unit interval via piecewise projective homemorphisms. A result of Lodha shows that $G$ in fact has type $F_{\infty}$. Here we will describe $G$ as a group that is "locally determined" by an inverse semigroup $S_{2}$, in the sense of the author's joint work with Hughes. The semigroup $S_{2}$ is generated by three linear fractional transformations $A$, $B$, and $C_{2}$, where $A$ and $B$ are elliptical transformations of the hyperbolic plane and $C_{2}$ is a hyperbo…

About von Neumann's problem for locally compact groups

We note a generalization of Whyte's geometric solution to the von Neumann problem for locally compact groups in terms of Borel and clopen piecewise translations. This strengthens a result of Paterson on the existence of Borel paradoxical decompositions for non-amenable locally compact groups. Along the way, we study the connection between some geometric properties of coarse spaces and certain algebraic characteristics of their wobbling groups.