A characterization of isometries of CAT(0)-space as maps preserving diagonal tube

CAT(0)-空间等距群的余循环的边界映射与可约性

Boundary maps and reducibility for cocycles into the isometries of CAT(0)-spaces

本文研究离散群在CAT(0)-空间可测场上的等距作用，证明无不变欧几里得子场时延拓到边界存在不变截面，并由此得到极大余循环的有限可约性与无穷维刚性现象。

Quasicircle boundaries and exotic almost-isometries

We consider properly discontinuous, isometric, convex cocompact actions of surface groups on a CAT(-1) space. We show that the limit set of such an action, equipped with the canonical visual metric, is a (weak) quasicircle in the sense of Falconer and Marsh. It follows that the visual metrics on such limit sets are classified, up to bi-Lipschitz equivalence, by their Hausdorff dimension. This result applies in particular to boundaries at infinity of the universal cover of a locally CAT(-1) surface. We show that any two periodic CAT(-1) metrics on $\mathbb H^2$ can be scaled so as to be almost-…

Topological mixing in CAT(-1) metric spaces

If X is a proper CAT(-1)-space and $Γ$ a non-elementary discrete group of isometries acting properly discontinuously on X, it is shown that the geodesic flow on the quotient space Y=X/$Γ$ is topologically mixing, provided that the generalized Busemann function has zeros on the boundary $\partial X$ and the non-wandering set of the flow equals the whole quotient space of geodesics GY:=GX/$Γ$ (the latter being redundant when Y is compact). Applications include the proof of topological mixing for (A) compact negatively curved polyhedra, (B) compact quotients of proper geodesically complete CAT(-1…

Existence and non-existence of bounded packing in CAT(0) spaces and Gromov hyperbolic spaces

The main result of this paper is that given a group $G$ acting geometrically by isometries on a CAT(0) space $X$ and a cyclic subgroup $H$ of $G$ generated by a rank-1 isometry of $X$, $H$ has bounded packing in $G$. We give two proofs of this result. The first one is by a characterization of rank-$1$ isometries by Hamenstadt. The second proof follows directly from some results of Dahmani-Guirardel-Osin and Sisto. Then using Mihailova's construction, we show the existence of a finitely generated subgroup of the direct product of two free groups $\mathbb F_2\times \mathbb F_2$ without the bound…

A strong Schottky lemma on $n$ generators for $\mathrm{CAT}(0)$ spaces

We give a criterion for a set of $n$ hyperbolic isometries of a $\mathrm{CAT}(0)$ metric space $X$ to generate a free group on $n$ generators. This extends a result by Alperin, Farb and Noskov who proved this for 2 generators under the additional assumption that $X$ is complete and has no fake zero angles. Moreover, when $X$ is locally compact, the group we obtain is also discrete.

Actions on products of CAT(-1) spaces

We show that for $X$ a proper $\mathrm{CAT}(-1)$ space there is a maximal open subset of the horofunction compactification of $X\times X$ with respect to the maximum metric that compactifies the diagonal action of an infinite quasi-convex group of the isometries of $X$. We also consider the product action of two quasi-convex representations of an infinite hyperbolic group on the product of two different proper $\mathrm{CAT}(-1)$ spaces.

CAT(0)空间上的随机游走与收缩元素

Marches aléatoires et éléments contractants sur des espaces CAT(0)

本论文研究CAT(0)空间上群作用的随机游走，利用收缩元素和双曲模型证明几乎必然边界收敛、漂移正性及中心极限定理等极限律，并证明仿射建筑上非初等等距作用总存在双曲等距映射。

On the dimension of CAT(0) spaces where mapping class groups act

Whenever the mapping class group of a closed orientable surface of genus g acts by semisimple isometries on a complete CAT(0) space of dimension less than g it fixes a point.

可见性空间的Tits二择性

The Tits alternative for visibility spaces

该研究证明了在满足有界填充性质的可见性CAT(0)空间上，以等距真不连续作用的有限生成群满足Tits二择性，并借此推导出此类作用下的有限生成挠群必为有限群且有全局不动点。

The rhombic dodecahedron and semisimple actions of Aut(F_n) on CAT(0) spaces

We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT$(0)$ spaces. If $n\ge 4$ then each of the Nielsen generators of Aut$(F_n)$ has a fixed point. If $n=3$ then either each of the Nielsen generators has a fixed point, or else they are hyperbolic and each Nielsen-generated $\Z^4\subset Aut(F_3)$ leaves invariant an isometrically embedded copy of Euclidean 3-space on which it acts as a discrete group of translations with the rhombic dodecahedron as a fundamental domain. An abundance of actions of the second kind is described. Constraints on maps from…

On the semi-direct product structure of CAT(0) groups

In this paper, we investigate finitely generated groups of isometries of CAT(0) spaces containing some central hyperbolic isometry, and study CAT(0) groups. We show that every CAT(0) group $Γ$ has the semi-direct product structure $Γ=(\cdots(((Γ'\rtimes\langleδ_{n}\rangle)\rtimes\langleδ_{n-1}\rangle)\rtimes\langleδ_{n-2}\rangle)\cdots)\rtimes\langleδ_{1}\rangle$ where $Γ'$ is a CAT(0) group with finite center and $δ_i\in Γ$ for $i=1,\dots,n$, and $Γ$ contains a finite-index subgroup $Γ'\times A$ where $A$ is isomorphic to ${\mathbb{Z}}^n$. We introduce some examples and remarks. Also we provi…

Sublinearly Morse Boundary I: CAT(0) Spaces

To every Gromov hyperbolic space X one can associate a space at infinity called the Gromov boundary of X. Gromov showed that quasi-isometries of hyperbolic metric spaces induce homeomorphisms on their boundaries, thus giving rise to a well-defined notion of the boundary of a hyperbolic group. Croke and Kleiner showed that the visual boundary of non-positively curved (CAT(0)) groups is not well-defined, since quasi-isometric CAT(0) spaces can have non-homeomorphic boundaries. For any sublinear function $κ$, we consider a subset of the visual boundary called the $κ$-Morse boundary and show that …