Search

We show that cubulated hyperbolic groups with spherical boundary of dimension 3 or at least 5 are virtually fundamental groups of closed, orientable, aspherical manifolds, provided that there are sufficiently many quasi-convex, codimension-1 subgroups whose limit sets are locally flat subspheres. The proof is based on ideas used by Markovic in his work on Cannon's conjecture for cubulated hyperbolic groups with 2-sphere boundary., Comment: 26 pages, 1 figure, 1 appendix. Comments welcome

We provide a systematic description of the automorphism groups of specially cocompact CAT(0) cube complexes. We show that these groups are topologically finitely generated, present a method to explicitly obtain generating sets, and prove a dichotomy on their size. Furthermore, we show that, under some extra assumptions, the normal subgroup known as Aut^+ is simple, non-discrete, and tdlc. In particular, we obtain a new class of simple, non-discrete, tdlc groups that are accessible to further study. Finally, we study the relative size of Aut^+ in the automorphism group, providing a sufficient condition for its closure to be finite index and presenting a common example where it is not even cocompact., Comment: 77 pages, 7 figures, comments welcome

We explicitly determine the automorphism groups of all self-similar trees (a.k.a. trees with finitely many cone types). We show that any such automorphism group is a direct limit of certain finite products of finite symmetric groups, which are parametrized by a certain deterministic finite automaton., Comment: 37 pages, 3 figures; comments welcome

We show that the sublinearly Morse boundary of a CAT(0) cubical group with a factor system is well-defined up to homeomorphism with respect to the visual topology. The key tool used in the proof is a new topology on sublinearly Morse boundaries that is induced by group actions on hyperbolic spaces that are sufficiently nice, for example, largest acylindrical actions. Using the same techniques, we obtain a explicit description of this new topology on the sublinearly Morse boundary of any hierarchically hyperbolic group in terms of medians. Finally, we explicitly describe the sublinear Morse boundaries of graph manifolds using their actions on Bass-Serre trees., Comment: 49 pages, 3 figures; comments welcome

Let $X$ be a Hadamard manifold with pinched negative curvature $-b^2\leq\kappa\leq -1$. Suppose $\Sigma\subseteq X$ is a totally geodesic, codimension-1 submanifold and consider the geodesic flow $\Phi^\nu_t$ on $X$ generated by a unit normal vector field $\nu$ on $\Sigma$. We say the normal growth exponent of $\Sigma$ in $X$ is at most $\beta$ if \[ \lim_{t \rightarrow \pm \infty} \frac{ \Vert d \Phi_t^\nu \Vert_{\infty} }{ e^{\beta \vert t \vert}} < \infty, \] where $\Vert d \Phi_t^\nu \Vert_{\infty} $ is the supremum of the operator norm of $d \Phi_t^\nu $ over all points of $\Sigma$. We show that if $\Sigma$ is bi-Lipschitz to hyperbolic $n$-space $\mathbb{H}^n$ and the normal growth exponent is at most 1, then $X$ is bi-Lipschitz to $\mathbb{H}^{n+1}$. As an application, we prove that if $M$ is a closed, negatively curved $(n+1)$-manifold, and $N\subset M$ is a totally geodesic, codimension-1 submanifold that is bi-Lipschitz to a hyperbolic manifold and whose normal growth exponent is at most 1, then $\pi_1(M)$ is isomorphic to a lattice in $\text{Isom}(\mathbb{H}^{n+1})$. Finally, we show that the assumption on the normal growth exponent is necessary in dimensions at least 4., Comment: 19 pages, 1 figure. Comments welcome!

We provide sufficient conditions as to when a boundary component of a cocompact convex set in a CAT(0)-space is contractible. We then use this to study when the limit set of a quasi-convex, codimension one subgroup of a negatively curved manifold group is `wild' in the boundary. The proof is based on a notion of coarse upper curvature bounds in terms of barycenters and the careful study of interpolation in geodesic metric spaces., Comment: 34 pages, 1 figure. The paper has been completely rewritten. The main result is now generalised to CAT(0) spaces. A connection between lambda-barycenters and Lipschitz-extension properties has been added

We prove that the sublinearly Morse boundary of every known cubulated group continuously injects in the Gromov boundary of a certain hyperbolic graph. We also show that for all CAT(0) cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.

We show that every cross ratio preserving homeomorphism between boundaries of Hadamard manifolds extends to a continuous map, called circumcenter extension, provided that the manifolds satisfy certain visibility conditions. We show that this map is a rough isometry, whenever the manifolds admit cocompact group actions by isometries and we improve the quasi-isometry constants provided by Biswas in the case of CAT(-1)} spaces. Finally, we provide a sufficient condition for this map to be an isometry in the case of Hadamard surfaces., Comment: V2: Correction of a factor 2 in Theorem D. Comments welcome. 54 pages, 6 figures

We compare several topologies on the Morse boundary $\partial_M Y$ of a $\mathrm{CAT(0)}$ cube complex $Y$. In particular, we show that the two topologies introduced by Cashen and Mackay are not equal in general and provide a new description of one of them in the language of cube complexes. As a corollary, we obtain a new approach to tackle the question whether the visual topology induces a quasi-isometry-invariant topology on the Morse boundary. This leads to an obstruction to quasi-isometry-invariance in terms of the behaviour of geodesics under quasi-isometries., Comment: 27 pages, 4 figures, comments welcome

We introduce a $\mathbb{Z}$-valued cross ratio on Roller boundaries of ${\rm CAT(0)}$ cube complexes. We motivate its relevance by showing that every cross-ratio preserving bijection of Roller boundaries uniquely extends to a cubical isomorphism. Our results are strikingly general and even apply to infinite dimensional, locally infinite cube complexes with trivial automorphism group., Comment: 19 pages, 3 figures. Final version, to appear on Groups Geom Dyn

We study cross ratios from an axiomatic viewpoint and show that a space equipped with a cross ratios carries several notions of dimension. Specifically, we introduce notions of Hausdorff- and Nagata-dimension and prove that they are invariants of M\"obius spaces. This provides us with more direct methods of obtaining dimensions for boundaries of Gromov-hyperbolic spaces., Comment: Revised version: The paper has been restructured, some proofs expanded and various typos corrected

We study cross ratios from an axiomatic viewpoint, also known as the study of M\"obius spaces. We characterise cross ratios induced by quasi-metrics in terms of topological properties of their image. Furthermore, we generalise the notions of Cauchy-sequences and completeness to M\"obius spaces and prove the existence of a unique completion under an extra assumption that, again, can be expressed in terms of the image of the cross ratio., Comment: Revised version: The paper has been restructured and some proofs shortened. Various typos etc. corrected