It is well-known that $\mathrm{SL}_{n}(\mathbf{Q}_{p})$ acts without fixed points on an $(n-1)$-dimensional $\mathrm{CAT}(0)$ space (the affine building). We prove that $n-1$ is the smallest dimension of $\mathrm{CAT}(0)$ spaces on which matrix groups act without fixed points. Explicitly, let $R$ be an associative ring with identity and $E_{n}^{\prime }(R)$ the extended elementary subgroup. Any isometric action of $E_{n}^{\prime }(R)$ on a complete $\mathrm{CAT(0)}$ space $X^{d}$ of dimension $d<n-1$ has a fixed point. Similar results are discussed for automorphism groups of free groups. Furthermore, we prove that any action of $\mathrm{Aut}(F_{n}),n\geq 3,$ on a uniquely arcwise connected space by homeomorphisms has a fixed point.
We show that the translation length of any parabolic isometry on a complete semi-uniformly visible CAT(0) space is always zero. As a consequence, we will classify the isometries on visible CAT(0) spaces in terms of translation lengths. We will also show that the moduli space $\mathbb{M}(S_{g,n})$ of surface $S_{g,n}$ of $g$ genus with $n$ punctures admits no complete visible CAT(0) Riemannian metric if $3g+n\geq 5$, which answers the Brock-Farb-McMullen question in the visible case.
The main technical result of this paper is to characterize the contracting isometries of a CAT(0) cube complex without any assumption on its local finiteness. Afterwards, we introduce the combinatorial boundary of a CAT(0) cube complex, and we show that contracting isometries are strongly related to isolated points at infinity, when the complex is locally finite. This boundary turns out to appear naturally in the context of Guba and Sapir's diagram groups, and we apply our main criterion to determine precisely when an element of a diagram group induces a contracting isometry on the associated …
We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such actions exist on CAT(0) spaces of dimension 3. Another example is the fundamental group, G, of a complete, non-compact, complex hyperbolic manifold M with finite volume, of complex-dimension n > 1. The group G is acting on the universal cover of M, which is isometric to H^n_C. It …
We give positive answers for questions by Berestovskii. Namely, we prove that every bijection of locally compact geodesically complete and connected at infinity CAT(0)-space $X$ onto itself preserving some fixed distance or satellite relations is an isometry of this space. The proof of this theorem is based on another result stated by Berestovskii as a problem: the metric of the space $X$ may be recovered from its diagonal tube corresponding to an arbitrary number $r > 0$.
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-…
Let S be an orientable surface of finite type and let Mod(S) be its mapping class group. We consider actions of Mod(S) by semisimple isometries on complete CAT(0) spaces. If the genus of S is at least 3, then in any such action all Dehn twists act as elliptic isometries. The action of Mod(S) on the completion of Teichmüller space with the Weil-Petersson metric shows that there are interesting actions of this type. 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 must fix a point. The …
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…
In this paper, based on research on rank-one isometries by W.Ballmann and M.Brin and recent research on rank-one isometries of Coxeter groups by P.Caprace and K.Fujiwara, we study a topological fractal structure of boundaries of Coxeter groups. We also show that the limit-point set is dense in a boundary of a Coxeter group and introduce some remarks on boundaries of CAT(0) groups with rank-one isometries.
Let $X$ be a CAT(0) space, and $G$ a discrete cyclic group of isometries of $X$. We investigate the domain of discontinuity for the action of $G$ on the boundary $\partial X$.
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…
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.
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.
We investigate the Tits boundary of locally compact CAT(0) 2-complexes. In particular we show that away from the endpoints, a geodesic segment in the Tits boundary is the ideal boundary of an isometrically embedded Euclidean sector. As applications, we provide sufficient conditions for two points in the Tits boundary to be the endpoints of a geodesic in the 2-complex and for a group generated by two hyperbolic isometries to contain a free group. We also show that if two CAT(0) 2-complexes are quasi-isometric then the cores of their Tits boundaries are bi-Lipschitz.
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.
We study groups acting on CAT(0) square complexes. In particular we show if Y is a nonpositively curved (in the sense of A. D. Alexandrov) finite square complex and the vertex links of Y contain no simple loop consisting of five edges, then any subgroup of the fundamental group of Y either is virtually free abelian or contains a free group of rank two. In addition we discuss when a group generated by two hyperbolic isometries contains a free group of rank two and when two points in the ideal boundary of a CAT(0) 2-complex at Tits distance $π$ apart are the endpoints of a geodesic in the 2-comp…
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 …
In this paper, we study CAT(0) groups and Coxeter groups whose boundaries are scrambled sets. Suppose that a group $G$ acts geometrically (i.e. properly and cocompactly by isometries) on a CAT(0) space $X$. (Such group $G$ is called a {\it CAT(0) group}.) Then the group $G$ acts by homeomorphisms on the boundary $\partial X$ of $X$ and we can define a metric $d_{\partial X}$ on the boundary $\partial X$. The boundary $\partial X$ is called a {\it scrambled set} if for any $α,β\in\partial X$ with $α\neqβ$, (1) $\limsup\{d_{\partial X}(gα,gβ) | g\in G\}>0$ and (2) $\liminf\{d_{\partial X}(gα,gβ)…
We consider actions of automorphism groups of free groups by semisimple isometries on complete CAT$(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…
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…
We show that every group acting freely and vertex-transitively by isometries on a product of two regular trees of finite valence is boundary rigid. That means that every CAT(0) space that admits a geometric action of any such group has the visual boundary homeomorphic to a join of two copies of the Cantor set.
Let $X$ be a proper geodesic metric space and let $G$ be a group of isometries of $X$ which acts geometrically. Cordes constructed the Morse boundary of $X$ which generalizes the contracting boundary for CAT(0) spaces and the visual boundary for hyperbolic spaces. We characterize Morse elements in $G$ by their fixed points on the Morse boundary $\partial_MX$. The dynamics on the Morse boundary is very similar to that of a $δ$-hyperbolic space. In particular, we show that the action of $G$ on $\partial_MX$ is minimal if $G$ is not virtually cyclic. We also get a uniform convergence result on th…
Let $X$ be a proper CAT(0) space and let $G$ be a cocompact group of isometries of $X$ which acts properly discontinuously. Charney and Sultan constructed a quasi-isometry invariant boundary for proper CAT(0) spaces which they called the contracting boundary. The contracting boundary imitates the Gromov boundary for $δ$-hyperbolic spaces. We will make this comparison more precise by establishing some well known results for the Gromov boundary in the case of the contracting boundary. We show that the dynamics on the contracting boundary is very similar to that of a $δ$-hyperbolic group. In part…
We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of proper, maximal hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse boundaries of two such spaces each contain at least three points, then the spaces are quasi-isometric if and only if there exists a 2-stable, quasi-möbius homeomorphism between their Morse boundaries. Our result extends a recent result of Charney-Murray, who prove such a classification for CAT(0) groups, and is new for mapping class groups and the fundamental group…
We prove that the Poisson boundary of a random walk with finite entropy on a non-elementary hyperbolic group can be identified with its hyperbolic boundary, without assuming any moment condition on the measure. We also extend our method to groups with an action by isometries on a hyperbolic metric space containing a WPD element; this applies to a large class of non-hyperbolic groups such as relatively hyperbolic groups, mapping class groups, and groups acting on CAT(0) spaces.
We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterised amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries.
We study extreme values of group-indexed stable random fields for discrete groups $G$ acting geometrically on spaces $X$ in the following cases: 1) $G$ acts freely, properly discontinuously by isometries on a CAT(-1) space $X$, 2) $G$ is a lattice in a higher rank Lie group, acting on a symmetric space $X$, 3) $G$ is the mapping class group of a surface acting on its Teichmuller space. The connection between extreme values and the geometric action is mediated by the action of the group $G$ on its limit set equipped with the Patterson-Sullivan measure. Based on motivation from extreme value the…
This is the second of two papers but has been written so as to have minimal dependence on the first paper (which is also on this archive). Let G be a group and let M be a CAT(0) proper metric space (e.g. a simply connected complete Riemannian manifold of non-positive sectional curvature or a locally finite tree). Assume G is of type F_n (type F_1 is finitely generated, type F_2 is finitely presented etc.) The "boundary", bdM, of M at infinity has two customary topologies - the compact "cone" topology and the Tits metric topology. We associate with any isometric action of G on M two subsets of …
We study $2$-dimensional Artin groups of hyperbolic type from the viewpoint of measure equivalence, and establish rigidity theorems. We first prove that they are boundary amenable. So is every group acting discretely by simplicial isometries on a connected piecewise hyperbolic $\mathrm{CAT}(-1)$ simplicial complex with countably many simplices in finitely many isometry types, assuming that vertex stabilizers are boundary amenable. Consequently, they satisfy the Novikov conjecture. We then show that measure equivalent $2$-dimensional Artin groups of hyperbolic type have isomorphic fixed set gra…
Hierarchically hyperbolic spaces provide a common framework for studying mapping class groups of finite type surfaces, Teichmüller space, right-angled Artin groups, and many other cubical groups. Given such a space $\mathcal X$, we build a bordificationcompatible with the hierarchically hyperbolic structure. If $\mathcal X$ is proper, we get a compactification of $\mathcal X$; we also prove that our construction generalizes the Gromov boundary of a hyperbolic space. In our first main set of applications, we introduce a notion of geometrical finiteness for hierarchically hyperbolic subgroups of…
