Related papers: A Generic Margulis Number for Hyperbolic 3-Manifol…

We describe a class of genus 2 closed hyperbolic 3-manifolds of arbitrarily large volume.

We show that for every $n\geq 2$ and any $\epsilon>0$ there exists a compact hyperbolic $n$-manifold with a closed geodesic of length less than $\epsilon$. When $\epsilon$ is sufficiently small these manifolds are non-arithmetic, and they…

Let M be a closed hyperbolic 3-manifold. We show that the number of genus g surface subgroups of the fundamental group of M grows like g^{2g}.

Given a closed hyperbolic 3-manifold M of volume V, and a link L in M such that the complement M \ L is hyperbolic, we establish a bound for the systole length of M \ L in terms of V. This extends a result of Adams and Reid, who showed that…

By a work of Thurston, it is known that if a hyperbolic fibred $3$-manifold $M$ has Betti number greater than 1, then $M$ admits infinitely many distinct fibrations. For any fibration $\omega$ on a hyperbolic $3$-manifold $M$, the number of…

We show that every sequence of torsion-free arithmetic congruence lattices in $\mathrm{PGL}(2,\mathbb R)$ or $\mathrm{PGL}(2,\mathbb C)$ satisfies a strong quantitative version of the Limit Multiplicity property. We deduce that for $R>0$ in…

We show the manifolds at infinity of the complex hyperbolic triangle groups $\Delta_{3,4,4;\infty}$ and $\Delta_{3,4,6;\infty}$,are one-cusped hyperbolic 3-manifolds $m038$ and $s090$ in the Snappy Census respectively.That is,these two…

Let $M = H^3/\Gamma$ be a hyperbolic 3-manifold, where $\Gamma$ is a non-elementary Kleinian group. It is shown that the length spectrum of $M$ is of unbounded multiplicity.

In this paper, we prove the Bounded Height Conjecture which the author formulated in [2]. As a corollary, it follows that there are only a finite number of hyperbolic three manifolds of bounded volume and trace field degree.

We show that there is an upper bound on the injectivity radius of a hyperbolic 3-manifold in terms of the the number of generators of its fundamental group.

We show that the number of isometry classes of cusped hyperbolic $3$-manifolds that bound geometrically grows at least super-exponentially with their volume, both in the arithmetic and non-arithmetic settings.

We show that for a representation of the fundamental group of a triangulated closed 3-manifold (not necessarily hyperbolic) into $\PSL$ so that any edge loop has non-trivial image under the representation, there exist uncountably many…

We obtain some restrictions on the topology of infinite volume hyperbolic manifolds. In particular, for any n and any closed negatively curved manifold M of dimension greater than 2, only finitely many hyperbolic n-manifolds are total…

We establish two-sided bounds for the complexity of two infinite series of closed orientable 3-dimensional hyperbolic manifolds, the Lobell manifolds and the Fibonacci manifolds.

We prove that every closed oriented 3-manifold admits a hyperbolic cone-manifold structure with cone-angle arbitrarily close to 2pi.

We prove existence of thick geodesic triangulations of hyperbolic 3-manifolds and use this to prove existence of universal bounds on the principal curvatures of surfaces embedded in hyperbolic 3-manifolds.

We conjecture that for every dimension n not equal 3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n at most 4 and n=6 this conjecture follows…

We provide a detailed proof of the following folklore theorem: Let mu > 0 be a Margulis constant for 3-dimensional hyperbolic space. Then for any d>0 there exists a constant K>0, depending on mu and d, so that for any complete finite volume…

We show that hyperbolic 3-manifolds with finitely generated fundamental group are tame, that is the ends are products. We actually work in slightly greater generality with pinched negatively curved manifolds with hyperbolic cusps. This…

A fundamental way to study 3-manifolds is through the geometric lens, one of the most prominent geometries being the hyperbolic one. We focus on the computation of a complete hyperbolic structure on a connected orientable hyperbolic…