Related papers: Arithmeticity, superrigidity and totally geodesic …

Our main result is that for all sufficiently large $x_0>0$, the set of commensurability classes of arithmetic hyperbolic 2- or 3-orbifolds with fixed invariant trace field $k$ and systole bounded below by $x_0$ has density one within the…

We describe a natural strategy to enumerate compact hyperbolic 3-manifolds with geodesic boundary in increasing order of complexity. We show that the same strategy can be employed to analyze simultaneously compact manifolds and…

In this paper, we prove an equality which involves Reidemeister torsion, complex volume, and Zograf infinite product for hyperbolic 3-manifolds with cusps.

In this paper, we show the existence of smoothly embedded closed minimal surfaces in infinite volume hyperbolic $3$-manifolds except some special cases.

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 prove a tubular neighborhood theorem for an embedded complex geodesic surface in a complex hyperbolic 2-manifold where the width of the tube depends only on the Euler characteristic of the embedded surface. We give an explicit estimate…

In this article, we give explicit examples of infinitely many non-commensurable (non-arithmetic) hyperbolic $3$-manifolds admitting exactly $k$ totally geodesic surfaces for any positive integer $k$, answering a question of Bader, Fisher,…

We investigate a question of Cooper adjacent to the Virtual Haken Conjecture. Assuming certain conjectures in number theory, we show that there exist hyperbolic rational homology 3-spheres with arbitrarily large injectivity radius. These…

It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length…

We prove the existence of manifolds with almost maximal volume entropy which are not hyperbolic.

In this paper we show that bending a finite volume hyperbolic $d$-manifold $M$ along a totally geodesic hypersurface $\Sigma$ results in a properly convex projective structure on $M$ with finite volume. We also discuss various geometric…

Let $(M,g)$ be a complete $(n+1)$-dimensional Riemannian manifold with $2\leq n\leq 6$. Our main theorem generalizes the solution of S.-T. Yau's conjecture on the abundance of minimal surfaces and builds on a result of M. Gromov. Suppose…

In this paper, we classify completely hyperbolic 3-manifolds corresponding to geometric limits of Kleinian surface groups isomorphic to $\pi_1(S)$ for a finite-type hyperbolic surface $S$. In the first of the three main theorems, we…

In this paper we show that totally geodesic subspaces determine the commensurability class of a standard arithmetic hyperbolic $n$-orbifold, $n\ge 4$. Many of the results are more general and apply to locally symmetric spaces associated to…

Let W be a compact hyperbolic n-manifold with totally geodesic boundary. We prove that if n>3 then the holonomy representation of pi_1 (W) into the isometry group of hyperbolic n-space is infinitesimally rigid.

According to Mostow's celebrated rigidity theorem, the geometry of closed hyperbolic 3-manifolds is already determined by their topology. In particular, the volume of such manifolds is a topological invariant and, as such, has been…

For a hyperbolic surface S of finite type we consider the set A(S) of angles between closed geodesics on S. Our main result is that there are only finitely many rational multiples of \pi in A(S).

We obtain strong upper bounds for the Betti numbers of compact complex-hyperbolic manifolds. We use the unitary holonomy to improve the results given by the most direct application of the techniques of [DS17]. We also provide effective…

We show that any group that is hyperbolic relative to virtually nilpotent subgroups, and does not admit peripheral splittings, contains a quasi-isometrically embedded copy of the hyperbolic plane. In natural situations, the specific…

We supply a proof of the fact that a hyperbolic 3-manifold $M$ with finitely generated fundamental group and with no parabolics is topologically tame. This proves the Marden's conjecture. Our approach is to form an exhaustion $M_i$ of $M$…