Parabolic groups acting on one-dimensional compact spaces

On fixed point sets of distinguished collections for groups of parabolic characteristic

We determine the nature of the fixed point sets of groups of order p, acting on complexes of distinguished p-subgroups (those p-subgroups containing p-central elements in their centers). The case when G has parabolic characteristic p is analyzed in detail.

Peripheral structures of relatively hyperbolic groups

In this paper, we introduce and characterize a class of parabolically extended structures for relatively hyperbolic groups. A characterization of relative quasiconvexity with respect to parabolically extended structures is obtained using dynamical methods. Some applications are discussed. The class of groups acting geometrically finitely on Floyd boundaries turns out to be easily understood. However, we also show that Dunwoody's inaccessible group does not act geometrically finitely on its Floyd boundary.

The Structure of Integral Parabolic Subgroups of Orthogonal Groups

We determine the detailed structure of parabolic subgroups of orthogonal groups over $\mathbb{Z}$, and deduce the precise form of canonical boundary components in toroidal compactifications of orthogonal Shimura varieties.

Indicability, residual finiteness, and simple subquotients of groups acting on trees

We establish three independent results on groups acting on trees. The first implies that a compactly generated locally compact group which acts continuously on a locally finite tree with nilpotent local action and no global fixed point is virtually indicable; that is to say, it has a finite index subgroup which surjects onto $\mathbf{Z}$. The second ensures that irreducible cocompact lattices in a product of non-discrete locally compact groups such that one of the factors acts vertex-transitively on a tree with a nilpotent local action cannot be residually finite. This is derived from a genera…

A fixed point theorem for Lie groups acting on buildings and applications to Kac-Moody theory

We establish a fixed point property for a certain class of locally compact groups, including almost connected Lie groups and compact groups of finite abelian width, which act by simplicial isometries on finite rank buildings with measurable stabilisers of points. As an application, we deduce amongst other things that all topological one-parameter subgroups of a real or complex Kac-Moody group are obtained by exponentiating ad-locally finite elements of the corresponding Lie algebra.

Groups not acting on compact metric spaces by homeomorphisms

We show that the direct sum of uncountably many non-Abelian groups does not embed into the group of homeomorphisms of a compact metric space.

Non-supramenable groups acting on locally compact spaces

Supramenability of groups is characterised in terms of invariant measures on locally compact spaces. This opens the door to constructing interesting crossed product C*-algebras for non-supramenable groups. In particular, stable Kirchberg algebras in the UCT class are constructed using crossed products for both amenable and non-amenable groups.

Special $p$-groups acting on compact manifolds

Riera proved at arXiv:1412.6964 that the diffeomorphism group of particular compact manifolds are not Jordan by exhibiting subgroups isomorphic to extra-special $p$-groups of exponent $p$ for primes $p$ satisfying some conditions. Generalising the methods of that paper, we construct a compact connected smooth real manifold for every natural number $r$ whose diffeomorphism group contains not only every extra-special $p$-group, but also every special $p$-group of order $p^r$ independently of its exponent for every prime $p$. We obtain a similar statement about finite Heisenberg groups as well as…

nilpotent群作用于一维紧致流形的临界正则性

Critical regularity of nilpotent groups acting on one-dimensional compact manifolds

研究nilpotent群在闭区间或半开区间及圆上的忠实微分作用的临界正则性，通过代数方法推导出表达式，推广了先前成果。

Higher generating subgroups and Cohen-Macaulay complexes

We show how to find higher generating families of subgroups, in the sense of Abels and Holz, for groups acting on Cohen-Macaulay complexes. We apply this to groups with a BN-pair to prove higher generation by parabolic and Levi-subgroups and describe higher generating families of parabolic subgroups in Aut(F_n).

Locally compact groups acting on trees, the type I conjecture and non-amenable von Neumann algebras

We address the problem to characterise closed type I subgroups of the automorphism group of a tree. Even in the well-studied case of Burger-Mozes' universal groups, non-type I criteria were unknown. We prove that a huge class of groups acting properly on trees are not of type I. In the case of Burger-Mozes groups, this yields a complete classification of type I groups among them. Our key novelty is the use of von Neumann algebraic techniques to prove the stronger statement that the group von Neumann algebra of the groups under consideration is non-amenable.

Groups not acting on manifolds

In this article we collect a series of observations that constrain actions of many groups on compact manifolds. In particular, we show that "generic" finitely generated groups have no smooth volume preserving actions on compact manifolds while also producing many finitely presented, torsion free groups with the same property.

离散 (P)-闭群作用在树上

Discrete (P)-closed Groups Acting On Trees

本文基于局部作用图，给出了作用在树上的群为离散的充要条件，该群满足Tits独立性性质(P)。

Splittings and automorphisms of relatively hyperbolic groups

We study automorphisms of a relatively hyperbolic group G. When G is one-ended, we describe Out(G) using a preferred JSJ tree over subgroups that are virtually cyclic or parabolic. In particular, when G is toral relatively hyperbolic, Out(G) is virtually built out of mapping class groups and subgroups of GL_n(Z) fixing certain basis elements. When more general parabolic groups are allowed, these subgroups of GL_n(Z) have to be replaced by McCool groups: automorphisms of parabolic groups acting trivially (i.e. by conjugation) on certain subgroups. Given a malnormal quasiconvex subgroup P of a h…

Weyl群作用于紧Lie群极大环面时的基本多胞体

Fundamental polytope for the Weyl group acting on a maximal torus of a compact Lie group

本文研究了有限Weyl群在紧Lie群极大环面上的作用，将其转化为扩展仿射Weyl群在对偶环面子代数上的作用，利用极小权理论给出了基本多胞体的凸包与半空间交集描述及自包含证明，并推导了扩展Dynkin图自同构群的结构推论。

Group $C^*$-algebras of locally compact groups acting on trees

We study the group $C^*$-algebras $C^*_{L^{p+}}(G)$ - constructed from $L^p$-integrability properties of matrix coefficients of unitary representations - of locally compact groups $G$ acting on (semi-)homogeneous trees of sufficiently large degree. These group $C^*$-algebras lie between the universal and the reduced group $C^*$-algebra. By directly investigating these $L^p$-integrability properties, we first show that for every non-compact, closed subgroup $G$ of the automorphism group $\mathrm{Aut}(T)$ of a (semi-)homogeneous tree $T$ that acts transitively on the boundary $\partial T$ and ev…