Let $G$ be a countable, Gromov-hyperbolic group. We say that $H$ is hyperbolically embedded in $G$ if $G$ is relatively hyperbolic to {$H$} (in the strong sense). This definition …

Do you know any formula to determine hyperbolic distance between two oposite sides of a hyperbolic hexagon uniquly determined by the lengts $l_1, l_2, l_3$ of alternating sides.

Good morning everyone, I was wondering about the difference between manifolds carying a Riemanniann metric with negative sectionnal curvature and hyperbolic manifolds. I was told …

So, if I start with a full Dihedral group D2n to represent a regular, ideal polygon in the hyperbolic plane, then I remove an element (and any subsequently necessary elements so th …

Good afternoon everyone, I have a very general question about hyperbolic manifolds and their fundamental groups in high dimension(at least $4$). If the theory of surfaces and $3$ …

Suppose $Y$ is a pair of pants with a hyperbolic structure and $\gamma_i; i = 1, 2, 3$ are the geodesic boundaries of length $l_i; i=1, 2, 3$ respectively. Now consider a essential …

The group mentioned in the title, $\langle x,y,z|xyzx^{-1}y^{-1}z^{-1}=1\rangle$, is in between the torus fundamental group $\langle x,y|xyx^{-1}y^{-1}=1\rangle$ and the two-holed …

We know now that hyperbolic 3-manifolds virtually embed into right-angled Artin groups as quasiconvex subgroups. Also, quasiconvexity depends on the generating set. I have been co …

What can be said about a discrete finitely generated subgroup $G$ of $PSL(2,\mathbb C)$ whose nontrivial elements are parabolic? If $G$ is geometrically finite, one can show that $ …

Good morning, I'm trying to understand the following fact, that is stated in Gromov and Thurston's paper "Pinching constants for hyperbolic manifolds" : Let $M$ be a (at least) 3 …

Is it possible to construct the midpoint of a segment in the hyperbolic plane using the set square only? With the set square one can draw the line through the given two …

$SL_2(\Bbb Z)$ acts on ${\Bbb R}^2$ fixing set-wise ${\Bbb Z}^2$, so $SL_2(\Bbb Z)$ acts on ${\Bbb R}^2\setminus {\Bbb Z}^2$, and then on the universal covering space of ${\Bbb R} …

Given a positive real number $l$. Does there exist a closed hyperbolic surface $X$ so that injectivity radius not less than $l$?

Hi, everyone. I am interested in the dehn filling and Hyperbolic 3-manifold. Suppose M be an orientable compact 3-manifold with one torus boundary and int(M) admit a hyperbolic …

My main research has been in hyperbolic geometry and geometric group theory. I always thought that the only real "application" of my work was that the universe is a 3-manifold. Bu …