In this paper, we show some splitting theorems for CAT(0) spaces on which a product group acts geometrically and we obtain a splitting theorem for compact geodesic spaces of non-positive curvature. A CAT(0) group $Γ$ is said to be {\it rigid}, if $Γ$ determines the boundary up to homeomorphisms of a CAT(0) space on which $Γ$ acts geometrically. C.Croke and B.Kleiner have constructed a non-rigid CAT(0) group. As an application of the splitting theorems for CAT(0) spaces, we obtain that if $Γ_1$ and $Γ_2$ are rigid CAT(0) groups then so is $Γ_1\times Γ_2$.
We show how to derive hyperbolicity of the free factor complex of $F_N$ from the Handel-Mosher proof of hyperbolicity of the free splitting complex of $F_N$, thus obtaining an alternative proof of a theorem of Bestvina-Feighn. We also show that under the natural map $τ$ from the free splitting complex to free factor complex, a geodesic $[x,y]$ maps to a path that is uniformly Hausdorff-close to a geodesic $[τ(x),τ(y)]$ .
We show that, given any finite dimensional, connected, compact metric space Z, there exists a group G acting geometrically on two CAT(0) spaces X and Y, a G-equivariant quasi-isometry f from X to Y, and a geodesic ray c in X, such that the closure of f(c), instersected with the boundary of Y, is homeomorphic to Z. This characterizes all homeomorphism types of "geodesic boundary images" that arise in this manner.
We prove that in CAT(0) spaces a quasi-geodesic is Morse if and only if it is contracting. Specifically, in our main theorem we prove that for $γ$ a quasi-geodesic in a CAT(0) space X, the following four statements are equivalent: (i) $γ$ is Morse, (ii) $γ$ is (b,c)--contracting, (iii), $γ$ is strongly contracting, and (iv) in every asymptotic cone $X_ω,$ any two distinct points in the ultralimit $γ_ω$ are separated by a cutpoint. As a corollary, we provide a converse to the usual Morse stability lemma in the CAT(0) setting. In addition, as a warm up we include an alternative proof of the fact…
The main theorem of this paper classifies the quasi-geodesics in a Coxeter group that are tracked by geodesics. As corollaries, we show that if a Coxeter group acts geometrically on a CAT(0) space X then CAT(0) rays (and lines) are tracked by Cayley graph geodesics, all special subgroups of the Coxeter group are quasi-convex in X, and in Cayley graphs for Coxeter groups, elements of infinite order are tracked by geodesics.
We introduce the notion of k-lower divergence for geodesic rays in CAT(0) spaces. Building on the work of Charney and Sultan we give various characterizations of k-contracting geodesic rays using k-lower divergence and k-slim triangles. We also characterize k-contracting geodesic rays in CAT(0) cube complexes using sequences of well-separated hyperplanes.
We construct a family of finite 2-complexes whose universal covers are CAT(0) and have polynomial divergence of desired degree. This answers a question of Gersten, namely whether such CAT(0) complexes exist.
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…
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.
In this paper we study CAT(0) groups and their splittings as graphs of groups. For one-ended CAT(0) groups with isolated flats we prove a theorem characterizing exactly when the visual boundary is locally connected. This characterization depends on whether the group has a certain type of splitting over a virtually abelian subgroup. In the locally connected case, we describe the boundary as a tree of metric spaces in the sense of Świątkowski. A significant tool used in the proofs of the above results is a general convex splitting theorem for arbitrary CAT(0) groups. If a CAT(0) group splits as …
Behrstock and Druţu raised a question about the existence of Morse geodesics in $CAT(0)$ spaces with divergence function strictly greater than $r^n$ and strictly less than $r^{n+1}$, where $n$ is an integer greater than $1$. In this paper, we answer the question of Behrstock and Druţu by showing that for each real number $s\geq 2$, there is a $CAT(0)$ space $X$ with a proper and cocompact action of some finitely generated group such that $X$ contains a Morse bi-infinite geodesic with the divergence equivalent to $r^s$.
For all systolic groups we construct boundaries which are EZ--structures. This implies the Novikov conjecture for torsion--free systolic groups. The boundary is constructed via a system of distinguished geodesics in a systolic complex, which we prove to have coarsely similar properties to geodesics in CAT(0) spaces.
We show the mapping class group, CAT(0) groups, the fundamental groups of closed 3-manifolds, and certain relatively hyperbolic groups have a local-to-global property for Morse quasi-geodesics. This allows us to generalize combination theorems of Gitik for quasiconvex subgroups of hyperbolic groups to the stable subgroups of these groups. In the case of the mapping class group, this gives combination theorems for convex cocompact subgroups. We show a number of additional consequences of this local-to-global property, including a Cartan-Hadamard type theorem for detecting hyperbolicity locally …
When two free factors A and B of a free group F_n are in "general position" we define the projection of B to the splitting complex (alternatively, the complex of free factors) of A. We show that the projections satisfy properties analogous to subsurface projections introduced by Masur and Minsky. We use the subfactor projections to construct an action of Out(F_n) on a finite product of hyperbolic spaces where every automorphism with exponential growth acts with positive translation length. We also prove a version of the Bounded geodesic image theorem. In the appendix, we give a sketch of the p…
We explain why semistability of a one-ended proper CAT(0) space can be determined by the geodesic rays. This is applied to boundaries of CAT(0) groups.
Let $X$ be a proper CAT($0$) space and $G$ a cocompact group of isometries of $X$ without fixed point at infinity. We prove that if $\partial X$ contains an invariant subset of circumradius $π/2$, then $X$ contains a quasi-dense, closed convex subspace that splits as a product. Adding the assumption that the $G$-action on $X$ is properly discontinuous, we give more conditions that are equivalent to a product splitting. In particular, this occurs if $\partial X$ contains a proper nonempty, closed, invariant, $π$-convex set in $\partial X$; or if some nonempty closed, invariant set in $\partial …
In this paper we show that dynamical and counting results characteristic of negatively-curved Riemannian geometry, or more generally CAT(-1) or rank-one CAT(0) spaces, also hold for geometrically-finite strictly convex projective structures equipped with their Hilbert metric. More specifically, such structures admit a finite Sullivan measure; with respect to this measure, the Hilbert geodesic flow is strongly mixing, and orbits and primitive closed geodesics equidistribute, allowing us to asymptotically enumerate these objects.
We show that the sublinearly Morse directions in the visual boundary of a rank-1 CAT(0) space with a geometric group action are generic in several commonly studied senses of the word, namely with respect to Patterson-Sullivan measures and stationary measures for random walks. We deduce that the sublinearly Morse boundary is a model of the Poisson boundary for finitely supported random walks on groups acting geometrically on rank-1 CAT (0) spaces. We prove an analogous result for mapping class group actions on Teichmüller space. Our main technical tool is a criterion, valid in any unique geodes…
Let X be a locally compact geodesically complete CAT(0) space and G be a discrete group acting properly and cocompactly on X. We show that G contains an element acting as a hyperbolic isometry on each indecomposable de Rham factor of X. It follows that if X is a product of d factors, then G contains free abelian group of rank d.
