Related papers: Counting surface subgroups in cusped hyperbolic 3-…

We count the number of conjugacy classes of maximal, genus g, surface subroups in hyperbolic 3-manifold groups. For any closed hyperbolic 3-manifold, we show that there is an upper bound on this number which grows factorially with g. We…

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}.

If M is a closed simple 3-manifold whose fundamental group contains a genus-g surface group for some g>1, and if the dimension of H_1(M;Z_2) is at least max(3g-1,6), we show that M contains a closed, incompressible surface of genus at most…

In this paper, we will determine the topological types of hyperbolic 3-anifolds H^3/G such that G is a geometric limit of any algebraically convergent sequence of quasi-Fuchsian groups.

Given two automorphisms of a group $G$, one is interested in knowing whether they are conjugate in the automorphism group of $G$, or in the abstract commensurator of $G$, and how these two properties may differ. When $G$ is the fundamental…

Almost-Fuchsian manifold is a class of complete hyperbolic three manifolds. Such a three-manifold is a quasi-Fuchsian manifold which contains a closed incompressible minimal surface with principal curvatures everywhere in the range of (-1,…

We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-1 surface subgroups has $\partial G \cong \mathbb{S}^2$. Combined with a result of Markovic, our result gives a new characterization of…

Let X be a hyperbolic surface and H the fundamental group of a hyperbolic 3-manifold that fibers over the circle with fiber X. Using the Birman exact sequence, H embeds in the mapping class group Mod(Y) of the surface Y obtained by removing…

A group $G$ is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of $G$, there exists a finite quotient of $G$ where the images of these subgroups are not conjugate. It is proved that the…

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 obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of $n$-generated one-ended…

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…

Let $M$ be an orientable 3-manifold with $\partial M$ a single torus. We show that the number of boundary slopes of immersed essential surfaces with genus at most $g$ is bounded by a quadratic function of $g$. In the hyperbolic case, this…

We establish a criterion for certain mapping classes of a surface homeomorphisms to be pseudo-Anosov in terms of the geometry of hyperbolic 3-manifolds and Gromov-hyperbolic surface group extensions. Specifically, any element of the…

This paper contains a purely topological theorem and a geometric application. The topological theorem states that if M is a simple closed orientable 3-manifold such that \pi_1(M) contains a genus g surface group and H_1(M;Z/2Z) has rank at…

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.

It is proved by Sakuma and Brooks that any closed orientable $3$-manifold with a Heegaard splitting of genus $g$ admits a $2$-fold branched cover that is a hyperbolic $3$-manifold and a genus $g$ surface bundle over the circle. This paper…

This paper presents some finiteness results for the number of boundary slopes of immersed essential surfaces of given genus g in a compact 3-manifold with torus boundary. In the case of hyperbolic 3-manifolds we obtain uniform quadratic…

Since the set of volumes of hyperbolic 3-manifolds is well ordered, for each fixed g there is a genus-g surface bundle over the circle of minimal volume. Here, we introduce an explicit family of genus-g bundles which we conjecture are the…

In this paper, we want to control the geometry of some surface subgroups of a cocompact Kleinian group. More precisely, provided any genus-2 quasi-Fuchsian group $\Gamma$ and cocompact Kleinian group $G$, then for any $K>1$, we will find a…