Related papers: Hyperbolic groups and spherical minimal surfaces

A group $\Gamma$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $\Gamma$ admits a cusp-uniform action on a proper $\delta$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric,…

We discuss how the global geometry and topology of manifolds depend on different group actions of their fundamental groups, and in particular, how properties of a non-trivial compact 4-dimensional cobordism $M$ whose interior has a complete…

Anosov representations $\rho$ of a hyperbolic group $\Gamma$ into a semisimple Lie group $G$ are known to admit cocompact domains of discontinuity in flag varieties $G/Q$, endowing the compact quotient manifolds $M_\rho$ with a…

In this paper, we classify all the orientable hyperbolic 5-manifolds that arise as a hyperbolic space form $H^5/\Gamma$ where $\Gamma$ is a torsion-free subgroup of minimal index of the congruence two subgroup $\Gamma^5_2$ of the group…

We give a generalization to convex co-compact semigroups of a beautiful theorem of Patterson-Sullivan, telling that the critical exponent (that is the exponential growth rate) equals the Hausdorff dimension of the limit set (that is the…

Let $\Gamma$ be a word hyperbolic group with a cyclic JSJ decomposition that has only rigid vertex groups, which are all fundamental groups of closed surface groups. We show that any group $H$ quasi-isometric to $\Gamma$ is abstractly…

A well known question of Gromov asks whether every one-ended hyperbolic group $\Gamma$ has a surface subgroup. We give a positive answer when $\Gamma$ is the fundamental group of a graph of free groups with cyclic edge groups. As a result,…

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…

We work with $N-$dimensional compact real hyperbolic space $X_{\Gamma}$ with universal covering $M$ and fundamental group $\Gamma$. Therefore, $M$ is the symmetric space $G/K$, where $G=SO_1(N,1)$ and $K=SO(N)$ is a maximal compact subgroup…

On any closed hyperbolizable 3-manifold, we find a sharp relation between the minimal surface entropy (introduced by Calegari-Marques-Neves) and the average area ratio (introduced by Gromov), and we show that, among metrics g with scalar…

For $n\geq 2$, let $\Gamma\subset \mathrm{SU}((n,1),\mathcal{O}_{K})$ be a torsion-free, finite-index subgroup, where $\mathcal{O}_K$ denotes the ring of integers of a totally imaginary number field $K$ of degree $2$. Let $\mathbb{B}^n$…

We show that the 1-cusped quotient of the hyperbolic space $\mathbb{H}^3$ by the tetrahedral Coxeter group $\Gamma_*=[5,3,6]$ has minimal volume among all non-arithmetic cusped hyperbolic 3-orbifolds, and as such it is uniquely determined.…

We investigate properties of the pseudo-Riemannian volume, entropy, and diameter for convex cocompact representations $\rho : \Gamma \to \mathrm{SO}(p,q+1)$ of closed $p$-manifold groups. In particular: We provide a uniform lower bound of…

Let P be the right-angled dodecahedron or 120-cell in hyperbolic space, and let W be the group generated by reflections across codimension-one faces of P. We prove that if Gamma is a torsion-free subgroup of minimal index in W, then the…

Let M be a hyperbolic n-manifold whose cusps have torus cross-sections. In arXiv:0901.0056, the authors constructed a variety of nonpositively and negatively curved spaces as "2\pi-fillings" of M by replacing the cusps of M with compact…

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,…

Let $\Gamma$ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $\Gamma$ into $G$, which generalizes the volume invariant for…

The conformal boundary of a hyperbolic $3$-manifold $M$ is a union of Riemann surfaces. If any of these Riemann surfaces has a nontrivial Teichm\"uller space, then the hyperbolic metric of $M$ can be deformed quasi-isometrically. These…

For closed and oriented hyperbolic surfaces, a formula of Witten establishes an equality between two volume forms on the space of representations of the surface in a semisimple Lie group. One of the forms is a Reidemeister torsion, the…

Let G be a word-hyperbolic group, obtained as a graph of free groups amalgamated along cyclic subgroups. If H_2(G;Q) is nonzero, then G contains a closed hyperbolic surface subgroup. Moreover, the unit ball of the Gromov-Thurston norm on…