Related papers: Commensurability and locally free Kleinian groups

Let N be a complete hyperbolic 3-manifold that is an algebraic limit of geometrically finite hyperbolic 3-manifolds. We show N is homeomorphic to the interior of a compact 3-manifold, or tame, if one of the following conditions holds: (1) N…

This is a survey of higher-dimensional Kleinian groups, i.e., discrete isometry groups of the hyperbolic n-space for n greater than 3. Our main emphasis is on the topological and geometric aspects of higher-dimensional Kleinian groups and…

By using Thurston's bending construction we obtain a sequence of faithful discrete representations \rho _n of the fundamental group of a closed hyperbolic 3-manifold fibering over the circle into the isometry group Iso H^4 of the hyperbolic…

We study a generalization of the Fuchsian triangle groups to the hyperbolic 3-space, namely, the groups generated by half-turns in three hyperbolic lines. The role of the hyperbolic triangles is now played by the right-angled hexagons. This…

In this paper we determine all Kobayashi-hyperbolic 2-dimensional complex manifolds for which the group of holomorphic automorphisms has dimension 3. This work concludes a recent series of papers by the author on the classification of…

We show that the mapping torus of a hyperbolic group by a hyperbolic automorphism is cubulable. Along the way, we (i) give an alternate proof of Hagen and Wise's theorem that hyperbolic free-by-cyclic groups are cubulable, and (ii) extend…

We prove that the abstract commensurator of a nonabelian free group, an infinite surface group, or more generally of a group that splits appropriately over a cyclic subgroup, is not finitely generated. This applies in particular to all…

In this paper we examine the geometry of minimal surfaces of arithmetic hyperbolic 3-manifolds. In particular, we give bounds on the totally geodesic 2-systole, construct infinitely many incommensurable manifolds with the same initial…

We prove hyperbolic 3-manifolds are geometrically inflexible: a unit quasiconformal deformation of a Kleinian group extends to an equivariant bi-Lipschitz diffeomorphism between quotients whose pointwise bi-Lipschitz constant decays…

In a variety of settings we provide a method for decomposing a 3-manifold $M$ into pieces. When the pieces have the appropriate type of hyperbolicity, then the manifold $M$ is hyperbolic and its volume is bounded below by the sum of the…

We prove the existence of Cannon-Thurston maps for Kleinian groups corresponding to pared manifolds whose boundary is incompressible away from cusps. We also describe the structure of these maps in terms of ending laminations.

The main thrust of present note is a volume formula for hyperbolic surface bundle with the fundamental group G. The novelty consists in a purely algebraic approach to the above problem. Initially, we concentrate on the Baum-Connes morphism…

In this note, we provide a description of the structure of homomorphisms from a finitely generated group to any torsion-free (3-dimensional) Kleinian group with uniformly bounded finite covolume. This is analogous to the Jorgensen-Thurston…

Let $N$ be a complete finite-volume hyperbolic $n$-manifold. An efficient cycle for $N$ is the limit (in an appropriate measure space) of a sequence of fundamental cycles whose $\ell^1$-norm converges to the simplicial volume of $N$. Gromov…

The aim of this paper is to demonstrate that very many Dehn fillings on a cusped hyperbolic 3-manifold yield a 3-manifold which is irreducible, atoroidal and not Seifert fibred, and which has infinite, word hyperbolic fundamental group. We…

The $\log 3$ Theorem, proved by Culler and Shalen, states that every point in the hyperbolic 3-space is moved a distance at least $\log 3$ by one of the non-commuting isometries $\xi$ or $\eta$ provided that $\xi$ and $\eta$ generate a…

We investigate the geometry of closed, orientable, hyperbolic $3$-manifolds whose fundamental groups are $k$-free for a given integer $k\ge 3$. We show that any such manifold $M$ contains a point $P$ of $M$ with the following property: If…

We show that there are at most finitely many one cusped orientable hyperbolic 3-manifolds which have more than eight non-hyperbolic Dehn fillings. Moreover, we show that determining these finitely many manifolds is decidable.

We show that any closed hyperbolic $3$-manifold $M$ has a co-final tower of covers $M_i \to M$ of degrees $n_i$ such that any subgroup of $\pi_1(M_i)$ generated by $k_i$ elements is free, where $k_i \ge n_i^C$ and $C = C(M) > 0$. Together…

This paper gives an exposition of the authors' harmonic deformation theory for 3-dimensional hyperbolic cone-manifolds. We discuss topological applications to hyperbolic Dehn surgery as well as recent applications to Kleinian group theory.…