Simplicity of the automorphism groups of order and tournament expansions of homogeneous structures

Automorphism groups of linearly ordered homogeneous structures

We apply results proved in [Li19] to the linear order expansions of non-trivial free homogeneous structures and the universal n-linear order for $n\geq 2$, and prove the simplicity of their automorphism groups.

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.

Non-universality of automorphism groups of uncountable ultrahomogeneous structures

In \cite{MbPe}, Mbombo and Pestov prove that the group of isometries of the generalized Urysohn space of density $κ$, for uncountable $κ$ such that $κ^{<κ}=κ$, is not a universal topological group of weight $κ$. We investigate automorphism groups of other uncountable ultrahomogeneous structures and prove that they are rarely universal topological groups for the corresponding classes. Our list of uncountable ultrahomogeneous structures includes random uncountable graph, tournament, linear order, partial order, group. That is in contrast with similar results obtained for automorphism groups of c…

余紧 CAT(0) 立方体复形的自同构群与单性

Automorphism groups of cocompact CAT(0) cube complexes and simplicity

本文研究特殊余紧 CAT(0) 立方体复形的自同构群，给出其生成集与大小二分法，并在额外条件下证明 Aut^+ 为单、非离散 tdlc 子群。

通过稳定子群确定超同质结构的自同构群的正规子群

Determining the normal subgroups of the automorphism groups of ultrahomogeneous structures via stabilisers

本文通过稳定子群确定超同质定向图的自同构群的正规子群，证明了通用n-超tournament和半通用tournament的自同构群的简单性，并给出了密集2π/n-局部顺序S(n)的自同构群简单性的新证明。

Automorphism groups of homogeneous structures with stationary weak independence relations

We generalise the result of Tent and Ziegler to homogeneous structures that have a stationary independence relation without the symmetry axiom. We apply our result to prove simplicity of the automorphism group of some asymmetric examples due to Cherlin.

Automorphism Groups of Trees: Generalities and Prescribed Local Actions

This article is an expanded version of the talks given by the authors at the Arbeitsgemeinschaft "Totally Disconnected Groups", held at Oberwolfach in October 2014. We recall the basic theory of automorphisms of trees and Tits' simplicity theorem, and present two constructions of tree groups via local actions with their basic properties: the universal group associated to a finite permutation group by M. Burger and S. Mozes, and the $k$-closures of a given group by C. Banks, M. Elder and G. Willis.

Boundedly simple groups of automorphisms of trees

A group is boundedly simple if, for some constant N, every nontrivial conjugacy class generates the whole group in N steps. For a large class of trees, Tits proved simplicity of a canonical subgroup of the automorphism group, which is generated by pointwise stabilizers of edges. We prove that only for uniform subdivisions of biregular trees are such groups boundedly simple. In fact these groups are 8-boundedly simple. As a consequence, we prove that if G is boundedly simple (or from a certain class K) and G acts by automorphisms on a tree, then G fixes some vertex of A, or stabilizes some end …

容许正则锦标赛 m-半正则表示的有限群

Finite groups admitting a regular tournament $m$-semiregular representation

本文针对 Du 提出的分类问题，证明了任意奇数阶大于 1 的有限群对所有奇数 m≥3 都容许正则锦标赛 m-半正则表示，从而完整回答了该问题。

On the acylindrical hyperbolicity of the tame automorphism group of $\mathrm{SL}_2(\mathbb{C})$

We introduce the notion of über-contracting element, a strengthening of the notion of strongly contracting element which yields a particularly tractable criterion to show the acylindrical hyperbolicity, and thus a strong form of non-simplicity, of groups acting on non locally compact spaces of arbitrary dimension. We also give a simple local criterion to construct über-contracting elements for groups acting on complexes with unbounded links of vertices. As an application, we show the acylindrical hyperbolicity of the tame automorphism group of $\mathrm{SL}_2(\mathbb{C})$, a subgroup of the $3$…

On the isometry group of the Urysohn space

We give a general criterion for the (bounded) simplicity of the automorphism groups of certain countable structures and apply it to show that the isometry group of the Urysohn space modulo the normal subgroup of bounded isometries is a simple group.

Simplicity and superrigidity of twin building lattices

Kac-Moody groups over finite fields are finitely generated groups. Most of them can naturally be viewed as irreducible lattices in products of two closed automorphism groups of non-positively curved twinned buildings: those are the most important (but not the only) examples of twin building lattices. We prove that these lattices are simple if and only if the corresponding buildings are (irreducible and) not of affine type (i.e. they are not Bruhat-Tits buildings). In fact, many of them are finitely presented and enjoy property (T). Our arguments explain geometrically why simplicity fails to ho…

Abstract simplicity of complete Kac-Moody groups over finite fields

Let $G$ be a Kac-Moody group over a finite field corresponding to a generalized Cartan matrix $A$, as constructed by Tits. It is known that $G$ admits the structure of a BN-pair, and acts on its corresponding building. We study the complete Kac-Moody group $\hat{G}$ which is defined to be the closure of $G$ in the automorphism group of its building. Our main goal is to determine when complete Kac-Moody groups are abstractly simple, that is have no proper non-trivial normal subgroups. Abstract simplicity of $\hat{G}$ was previously known to hold when A is of affine type. We extend this result t…

On regular CAT(0) cube complexes

We provide a necessary and sufficient condition on a finite flag simplicial complex, L, for which there exists a unique CAT(0) cube complex whose vertex links are all isomorphic to L. We then find new examples of such CAT(0) cube complexes and prove that their automorphism groups are virtually simple. The latter uses a result, which we prove in the appendix, about the simplicity of certain subgroups of the automorphism group of a rank-one CAT(0) cube complex. This result generalizes previous results by Tits and by Haglund and Paulin.

On free products of graphs

We define a free product of connected simple graphs that is equivalent to several existing definitions when the graphs are vertex-transitive but differs otherwise. The new definition is designed for the automorphism group of the free product to be as large as possible, and we give sufficient criteria for it to be non-discrete. Finally, we transfer Tits' classification of automorphisms of trees and simplicity criterion to free products of graphs.

康威-威尔士格与鲁德瓦利斯群及蒂茨群

The Conway-Wales lattice and the Rudvalis and Tits groups

本文从交织汉明码ℋ₇与ℋ₈的Z₄自对偶偶码构造康威-威尔士格，确定其自同构群，导出鲁德瓦利斯群与蒂茨群，相关证明均在格内完成。

C$^*$-simple groups: amalgamated free products, HNN extensions, and fundamental groups of 3-manifolds

We establish sufficient conditions for the C$^*$-simplicity of two classes of groups. The first class is that of groups acting on trees, such as amalgamated free products, HNN-extensions, and their normal subgroups; for example normal subgroups of Baumslag-Solitar groups. The second class is that of fundamental groups of compact 3-manifolds, related to the first class by their Kneser-Milnor and JSJ-decompositions. Much of our analysis deals with conditions on an action of a group $Γ$ on a tree $T$ which imply the following three properties: abundance of hyperbolic elements, better called stron…

C*-代数的酉群的李理论方法

Lie theoretic approach to unitary groups of $C^*$-algebras

本文利用 Banach-Lie 群理论研究单位 C*-代数的酉群与近似内自同构群，用完美理想刻画其闭正规子群结构，并建立局部 AF 代数情形下的一一对应，进而用酉群结构刻画代数简单性，推广了 Robert 的结果。