Related papers: Pinching surface groups in complex hyperbolic plan…

In this article, we prove that the commensurability class of a closed, orientable, hyperbolic 3-manifold is determined by the surface subgroups of its fundamental group. Moreover, we prove that there can be only finitely many closed,…

We construct triangular hyperbolic polyhedra whose links are generalized 4-gons. The universal cover of those polyhedra are hyperbolic buildings, which appartments are hyperbolic planes tesselated by regular triangles with angles $\pi/4$.…

We prove a division algorithm for group rings of high genus surface groups and use it to show that some $2$-complexes with surface fundamental groups are standard. We also give an application of division to cohomological dimension of…

We construct nonlinear hyperbolic groups which are large, torsion-free, one-ended, and admit a finite $K(\pi,1)$. Our examples are built from superrigid cocompact rank one lattices via amalgamated free products and HNN extensions.

In this paper, we construct geometrically finite rational maps with buried critical points on the boundaries of some hyperbolic components by using the pinching and plumbing deformations.

The study of geometric group theory has suggested several theorems related to subdivision tilings that have a natural hyperbolic structure. However, few examples exist. We construct subdivision tilings for the complement of every…

We classify surface Houghton groups, as well as their pure subgroups, up to isomorphism, commensurability, and quasi-isometry.

We show that surface groups are flexibly stable in permutations. This is the first non-trivial example of a non-amenable flexibly stable group. Our method is purely geometric and relies on an analysis of branched covers of hyperbolic…

This is an expository article about groups generated by two isometries of the complex hyperbolic plane.

We give examples of cohomological automorphic forms for unitary groups which are $p$-adically rigid.

We prove that the automorphism group of an affine, cubic surface with equation $xyz=g(x,y)$ contains ${\mathbb Z}$ as a finite index subgroup. These equations were first studied by Mordell. v.2: small changes, references updated.

Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying…

We study quartic surfaces that admit a group of projective automorphisms isomorphic to icosahedron group.

In this paper we give the asymptotic growth of the number of connected components of the moduli space of surfaces of general type corresponding to certain families of Beauville surfaces with group either $\PSL(2,p)$, or an alternating…

We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to…

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…

We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.

We construct explicit families of quasi-hyperbolic and hyperbolic surfaces. This is based on earlier work of Vojta, and the recent expansion and generalization of it by the first author. In this paper we further extend it to the singular…

Let $G$ denote the projective special linear group $\text{PSL}(2,q)$, for a prime power $q$. It is shown that a finite 2-subgroup of the group $V(\mathbb{Z}G)$ of augmentation 1 units in the integral group ring $\mathbb{Z}G$ of $G$ is…

Beauville surfaces are a class of complex surfaces defined by letting a finite group $G$ act on a product of Riemann surfaces. These surfaces possess many attractive geometric properties several of which are dictated by properties of the…