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Let $M$ be a $2$-cusped hyperbolic $3$-manifold. By the work of Thurston, the product of the derivatives of the holonomies of core geodesics of each Dehn filling of $M$ is an invariant of it. In this paper, we classify Dehn fillings of $M$…

We show that a complete hyperbolic n-manifold has a geodesic triangulation such that the tetrahedra contained in the thick part are L-bilipschitz diffeomorphic to the standard Euclidean n-simplex, for some constant L depending only on the…

Deformations of hyperbolic manifolds through metrics with cone singularities along closed loops were first studied by Thurston as continuous realisations of Dehn fillings. Instead of gluing singular solid tori into rank $2$ cusps, we glue…

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…

Thurston's hyperbolization theorem for Haken manifolds and normal surface theory yield an algorithm to determine whether or not a compact orientable 3-manifold with nonempty boundary consisting of tori admits a complete finite-volume…

It is a theorem of Casson and Rivin that the complete hyperbolic metric on a cusp end ideal triangulated 3-manifold maximizes volume in the space of all positive angle structures. We show that the conclusion still holds if some of the…

Given an ideal triangulation of a connected 3-manifold with non-empty boundary consisting of a disjoint union of tori, a point of the deformation variety is an assignment of complex numbers to the dihedral angles of the tetrahedra subject…

We establish a bijective correspondence between the set T(n) of 3-dimensional triangulations with n tetrahedra and a certain class H(n) of relative handlebodies (i.e. handlebodies with boundary loops, as defined by Johannson) of genus n+1.…

We give three infinite families of examples of nonhyperbolic Dehn fillings on hyperbolic manifolds. A manifold in the first family admits two Dehn fillings of distance two apart, one of which is toroidal and annular, and the other is…

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.

It is conjectured that every cusped hyperbolic 3-manifold admits a geometric triangulation, i.e. it is decomposed into positive volume ideal hyperbolic tetrahedra. Here, we show that sufficiently highly twisted knots admit a geometric…

We give a solution to Dehn's isomorphism problem for the class of all hyperbolic groups, possibly with torsion. We also prove a relative version for groups with peripheral structures. As a corollary, we give a uniform solution to…

Let $M$ be a volume finite non-compact complete hyperbolic $n$-manifold with totally geodesic boundary. We show that there exists a polyhedral decomposition of $M$ such that each cell is either an ideal polyhedron or a partially truncated…

The work of Jorgensen and Thurston shows that there is a finite number N(v) of orientable hyperbolic 3-manifolds with any given volume v. We show that there is an infinite sequence of closed orientable hyperbolic 3-manifolds, obtained by…

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.…

We consider in this paper the minimally twisted chain link with 5 components in the 3-sphere, and we analyze the Dehn surgeries on it, namely the Dehn fillings on its exterior M5. The 3-manifold M5 is a nicely symmetric hyperbolic one,…

We construct the first example of a ``one-cusped'' hyperbolic 3-orbifold for which we see the true shape of the space of hyperbolic Dehn fillings.

We consider non-orientable hyperbolic 3-manifolds of finite volume $M^3$. When $M^3$ has an ideal triangulation $\Delta$, we compute the deformation space of the pair $(M^3, \Delta)$ (its Neumann Zagier parameter space). We also determine…

Thurston's earthquake theorem asserts that every orientation-preserving homeomorphism of the circle admits an extension to the hyperbolic plane which is a (left or right) earthquake. The purpose of these notes is to provide a proof of…

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…