Related papers: Higher rank hyperbolicity

Given a domain $G \subsetneq \Rn$ we study the quasihyperbolic and the distance ratio metrics of $G$ and their connection to the corresponding metrics of a subdomain $D \subset G$. In each case, distances in the subdomain are always larger…

We investigate CAT(0) metric spaces whose associated Tits boundary is compact. Prominent examples of such spaces are of course the euclidean ones. However there exist non trivial geodesically complete CAT(0) spaces with compact Tits…

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a…

We present a new notion of non-positively curved groups: the collection of discrete countable groups acting (AU-)acylindrically on finite products of $\delta$-hyperbolic spaces with general type factors. Inspired by the classical theory of…

In this note, we investigate the existence of smooth complete hypersurfaces in hyperbolic space with constant $(n-2)$-curvature and a prescribed asymptotic boundary at infinity. Previously, the existence was known only for a restricted…

A CAT(0) space has rank at least $n$ if every geodesic lies in an $n$-flat. Ballmann's Higher Rank Rigidity Conjecture predicts that a CAT(0) space of rank at least $2$ with a geometric group action is rigid -- isometric to a Riemannian…

In this note we show that for any hyperbolic surface S, the number of geodesics of length bounded above by L in the mapping class group orbit of a fixed closed geodesic with a single double point is asymptotic to L raised to the dimension…

We study relations between maps between relatively hyperbolic groups/spaces and quasisymmetric embeddings between their boundaries. More specifically, we establish a correspondence between (not necessarily coarsely surjective)…

This paper studies sharp and rigid isoperimetric comparison theorems and asymptotic isoperimetric properties for small and large volumes on $N$-dimensional ${\rm RCD}(K,N)$ spaces $(X,\mathsf{d},\mathscr{H}^N)$. Moreover, we obtain almost…

Let N be the moduli space of stable rank 2 quasiparabolic vector bundles of fixed degree on the projective line with 2g+1 marked points, where g>1, and stability is with respect to the weights {0,1/2} at each marked point. In this note we…

Given an irreducible non-spherical non-affine (possibly non-proper) building $X$, we give sufficient conditions for a group $G < \Aut(X)$ to admit an infinite-dimensional space of non-trivial quasi-morphisms. The result applies to all…

We study those groups that act properly discontinuously, cocompactly, and isometrically on CAT(0) spaces with isolated flats and the Relative Fellow Traveller Property. The groups in question include word hyperbolic CAT(0) groups as well as…

We prove several results on the number of solutions to the asymptotic Plateau problem in $\mathbb H^3$. Firstly we discuss criteria that ensure uniqueness. Given a Jordan curve $\Lambda$ in the asymptotic boundary of $\mathbb H^3$, we show…

A family of non-radial solutions of the Yamabe equation, with reference the hyperbolic space, is constructed using power series. As a result, we obtain a family of asymptotically hyperbolic metrics, with spherical conformal infinity, with…

We introduce a class of mappings called vertical quasi-isometries and show that branched quasisymmetries $X\to Y$ of Guo and Williams between compact, bounded turning metric doubling spaces admit natural vertically quasi-isometric…

In this article we exhibit the largest constant in a quadratic isoperimetric inequality which ensures that a geodesic metric space is Gromov hyperbolic. As a particular consequence we obtain that Euclidean space is a borderline case for…

We provide an analysis of the dynamics of isometries and semicontractions of metric spaces. Certain subsets of the boundary at infinity play a fundamental role and are identified completely for the standard boundaries of CAT(0)-spaces,…

A famous open problem asks whether the asymptotic dimension of a CAT(0) group is necessarily finite. For hyperbolic groups, it is known that asymptotic dimension of the group is bounded above by the dimension of the boundary plus one, which…

Let $M$ be a closed hyperbolic $3$-manifold. A homotopy class $[S]$ of surfaces in $M$ is filling if any representative cuts $M$ into components contractible in $M$. We prove that there exist $\epsilon_0, g_0>0$ such that every homotopy…

Let $S$ be a rank-one symmetric space of non-compact type and let $X$ be a $\text{CAT}(-1)$ space. A well-known result by Bourdon states that if a topological embedding $\varphi: \partial_\infty S \rightarrow \partial_\infty X$ respects…