有限秩中位代数上作用的凸核

The Tits alternative for finite rank median spaces

We prove a version of the Tits alternative for groups acting on complete, finite rank median spaces. This shows that group actions on finite rank median spaces are much more restricted than actions on general median spaces. Along the way, we extend to median spaces the Caprace-Sageev machinery and part of Hagen's theory of unidirectional boundary sets.

Superrigidity of actions on finite rank median spaces

Finite rank median spaces are a simultaneous generalisation of finite dimensional ${\rm CAT}(0)$ cube complexes and real trees. If $Γ$ is an irreducible lattice in a product of rank one simple Lie groups, we show that every action of $Γ$ on a complete, finite rank median space has a global fixed point. This is in sharp contrast with the behaviour of actions on infinite rank median spaces. The fixed point property is obtained as corollary to a superrigidity result; the latter holds for irreducible lattices in arbitrary products of compactly generated groups. In previous work, we introduced "Rol…

粗高阶中位数、对称空间与凸射影几何

Coarse higher medians, symmetric spaces, and convex projective geometry

引入粗\(r -\)中位数空间概念，是Bowditch粗中位数空间的高阶类似物，在拟等距下稳定。利用凸射影几何工具，证明几类高阶对称空间存在粗\(r -\)中位数，给出诸多不具有Bowditch粗中位数的例子。

局部紧有限秩中值空间上的等距作用

Isometric actions on locally compact finite rank median spaces

该研究针对局部紧有限秩中值空间的等距作用问题，证明了容许传递作用的此类空间等距于带ℓ¹度量的ℝⁿ，还在作用正则性条件下得到轨道离散性，并给出完备有限秩中值空间紧性的半空间组合刻画。

Roller boundaries for median spaces and algebras

We construct compactifications for median spaces with compact intervals, generalising Roller boundaries of ${\rm CAT}(0)$ cube complexes. Examples of median spaces with compact intervals include all finite rank median spaces and all proper median spaces of infinite rank. Our methods also work for general median algebras, where we recover the zero-completions of Bandelt and Meletiou. Along the way, we prove various properties of halfspaces in finite rank median spaces and a duality result for locally convex median spaces.

Limit sets for convex cocompact groups in higher rank symmetric spaces

We show that every limit point of a Zariski dense discrete subgroup $Γ$ of the isometry group of a symmetric space of noncompact type is conical if and only if $Γ$ is convex cocompact.