Two-sided bounds for the complexity of cyclic branched coverings of two-bridge links

We consider closed orientable 3-dimensional hyperbolic manifolds which are cyclic branched coverings of the 3-sphere, with branching set being a two-bridge knot (or link). We establish two-sided linear bounds depending on the order of the covering for the Matveev complexity of the covering manifold. The lower estimate uses the hyperbolic volume and results of Cao-Meyerhoff and Gueritaud-Futer (who recently improved previous work of Lackenby), while the upper estimate is based on an explicit triangulation, which also allows us to give a bound on the Delzant T-invariant of the fundamental group of the manifold.Comment: Estimates improved using recent results of Gueritaud-Futer and Kim-Ki

Generalized Teichm\"{u}ller space of non-compact 3-manifolds and Mostow rigidity

Consider a 3$-$dimensional manifold $N$ obtained by gluing a finite number of ideal hyperbolic tetrahedra via isometries along their faces. By varying the isometry type of each tetrahedron but keeping fixed the gluing pattern we define a space $\mathcal{T}$ of complete hyperbolic metrics on $N$ with cone singularities along the edges of the tetrahedra. We prove that $\mathcal{T}$ is homeomorphic to a Euclidean space and we compute its dimension. By means of examples, we examine if the elements of $$ are uniquely determined by the angles around the edges of $N.$Comment: 15 pages, 7 figure

On volumes of hyperideal tetrahedra with constrained edge lengths

Hyperideal tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional topology, like the computation of quantum invariants of 3-manifolds and the use of variational methods in the study of circle packings on surfaces. The Schl\"afli formula neatly describes the behaviour of the volume of hyperideal tetrahedra with respect to dihedral angles, while the dependence of volume on edge lengths is worse understood. In this paper we prove that, for every $\ell<\ell_0$, where $\ell_0$ is an explicit constant, regular hyperideal tetrahedra of edge length $\ell$ maximize the volume among hyperideal tetrahedra whose edge lengths are all not smaller than $\ell$. This result provides a fundamental step in the computation of the ideal simplicial volume of an infinite family of hyperbolic 3-manifolds with geodesic boundary.Comment: 20 pages, 2 figures, Some minor changes, To appear in Periodica Mathematica Hungaric

On deformations of hyperbolic 3-manifolds with geodesic boundary

Let M be a complete finite-volume hyperbolic 3-manifold with compact non-empty geodesic boundary and k toric cusps, and let T be a geometric partially truncated triangulation of M. We show that the variety of solutions of consistency equations for T is a smooth manifold or real dimension 2k near the point representing the unique complete structure on M. As a consequence, the relation between deformations of triangulations and deformations of representations is completely understood, at least in a neighbourhood of the complete structure. This allows us to prove, for example, that small deformations of the complete triangulation affect the compact tetrahedra and the hyperbolic structure on the geodesic boundary only at the second order.Comment: This is the version published by Algebraic & Geometric Topology on 23 March 200

Dehn filling of cusped hyperbolic 3-manifolds with geodesic boundary

We define for each g>=2 and k>=0 a set M_{g,k} of orientable hyperbolic 3-manifolds with $k$ toric cusps and a connected totally geodesic boundary of genus g. Manifolds in M_{g,k} have Matveev complexity g+k and Heegaard genus g+1, and their homology, volume, and Turaev-Viro invariants depend only on g and k. In addition, they do not contain closed essential surfaces. The cardinality of M_{g,k} for a fixed k has growth type g^g. We completely describe the non-hyperbolic Dehn fillings of each M in M_{g,k}, showing that, on any cusp of any hyperbolic manifold obtained by partially filling M, there are precisely 6 non-hyperbolic Dehn fillings: three contain essential discs, and the other three contain essential annuli. This gives an infinite class of large hyperbolic manifolds (in the sense of Wu) with boundary-reducible and annular Dehn fillings having distance 2, and allows us to prove that the corresponding upper bound found by Wu is sharp. If M has one cusp only, the three boundary-reducible fillings are handlebodies.Comment: 28 pages, 16 figure