Related papers: Explicit equivalences between CAT(0) hyperbolic ty…

In this note, we present examples of non-quasi-geodesic metric spaces which are hyperbolic (i.e., satisfying the Gromov's $4$-point condition) while the intersection of any two metric balls therein does not either "look like" a ball or has…

We prove that if X is a complete geodesic metric space with uniformly generated first homology group and $f: X\to R$ is metrically proper on the connected components and bornologous, then X is quasi-isometric to a tree. Using this and…

Let G be a group acting geometrically on a CAT(0) cube complex X. We prove first that G is hyperbolic relative to the collection P of subgroups if and only if the simplicial boundary of X is the disjoint union of a nonempty discrete set,…

We examine Euclidean plane domains with their hyperbolic or quasihyperbolic distance. We prove that the associated metric spaces are quasisymmetrically equivalent if and only if they are bi-Lipschitz equivalent. On the other hand, for…

We prove that quasi-isometries of horospherical products of hyperbolic spaces are geometrically rigid in the sense that they are uniformly close to product maps, this is a generalisation of the result obtained by Eskin, Fisher and Whyte in…

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…

Let $\Gamma$ be a non-elementary Gromov-hyperbolic group, and $\partial \Gamma$ denote its Gromov boundary. We consider $\Gamma$-invariant proper $\delta$-hyperbolic, quasi-convex metric $d$ on $\Gamma$, and the associated…

The space of positively curved hermitian metrics on a positive holomorphic line bundle over a compact complex manifold is an infinite-dimensional symmetric space. It is shown by Phong and Sturm that geodesics in this space can be uniformly…

We generalize the notion of Gromov boundary to a larger class of metric spaces beyond Gromov hyperbolic spaces. Points in this boundary are classes of quasi-geodesic rays and the space is equipped with a topology that is naturally invariant…

The class of coarsely convex spaces is a coarse geometric analogue of the class of nonpositively curved Riemannian manifolds. It includes Gromov hyperbolic spaces, CAT(0) spaces, proper injective metric spaces and systolic complexes. It is…

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 $\Gamma$ is said to be {\it…

We prove prime geodesic theorems counting primitive closed geodesics on a compact hyperbolic 3-manifold with length and holonomy in prescribed intervals, which are allowed to shrink. Our results imply effective equidistribution of holonomy…

Exploiting a relationship between closed geodesics on a generic closed hyperbolic surface S and a certain unipotent flow on the product space T_1(S) x T_1(S), we obtain a local asymptotic equidistribution result for long closed geodesics on…

Let $f \in C^2(\mathbb{T}^2)$ have mean value 0 and consider $$ \sup_{\gamma~{\tiny \mbox{closed geodesic}}}{~~~ \frac{1}{|\gamma|} \left| \int_{\gamma}{ f ~~d\mathcal{H}^1}\right| },$$ where $\gamma$ ranges over all closed geodesics…

For every proper geodesic space $X$ we introduce its quasi-geometric boundary $\partial_{QG}X$ with the following properties: 1. Every geodesic ray $g$ in $X$ converges to a point of the boundary $\partial_{QG}X$ and for every point $p$ in…

This paper is a survey dedicated to the following question: given a group acting on some CAT(0) cube complex, how to exploit this action to determine whether or not the group is Gromov / relatively / acylindrically hyperbolic? As much as…

We use the interplay between combinatorial and coarse geometric versions of negative curvature to investigate the geometry of infinitely presented graphical $Gr'(1/6)$ small cancellation groups. In particular, we characterize their…

We analyze the coarse geometry of the Weil-Petersson metric on Teichm\"uller space, focusing on applications to its synthetic geometry (in particular the behavior of geodesics). We settle the question of the strong relative hyperbolicity of…

Let M be an orientable hyperbolic surface without boundary and let $\gamma$ be a closed geodesic in M. We prove that any side of any triangle formed by distinct lifts of $\gamma$ in H2 is shorter than $\gamma$.

Suppose G is a hyperbolic group whose boundary has topological dimension k. If the boundary is quasisymmetrically homeomorphic to an Ahlfors k-regular metric space, then, modulo a finite normal subgroup, G is isomorphic to a uniform lattice…