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153
Hyperbolic structures on 3–manifolds II: Surface groups and 3–manifolds which fiber over the circle
, 1998
"... The main result (0.1) of this paper is that every atoroidal threemanifold that fibers over the circle has a hyperbolic structure. Consequently, every fibered threemanifold admits a geometric decomposition. The main tool for constructing hyperbolic structures on fibered threemanifolds is the dou ..."
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Cited by 148 (2 self)
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The main result (0.1) of this paper is that every atoroidal threemanifold that fibers over the circle has a hyperbolic structure. Consequently, every fibered threemanifold admits a geometric decomposition. The main tool for constructing hyperbolic structures on fibered threemanifolds is the double limit theorem (4.1), which is of interest for its own sake and lays out general conditions under which sequences of quasiFuchsian groups have algebraically convergent subsequences. The main tool in proving the double limit theorem is an analysis of the geometry of hyperbolic manifolds that are homotopy equivalent to a surface. This analysis is also of interest in its own right. This eprint is based on the August 1986 version of this preprint, which was submitted, refereed, and accepted for publication; for reasons that are hard to fathom, I never returned a corrected version to the journal. I apologize for my long neglect of its publication, and I want to thank the referee for detailed comments which have been incorporated into the present eprint. No other significant changes have been made, except conversion to LATEX, which has resulted in changes in numbering. The 1986 preprint was in turn a revision of a 1981 preprint; the various versions were fairly widely circulated in the early 1980’s, and the results became widely known and used. Too many developments have intervened to be easily summarized, except for pointers particularly to the works of McMullen [McM96] and Otal [Ota96] that give alternative proofs for the main results of this paper and contain other interesting material as well.
Quantum gravity in 2 + 1 dimensions . . .
 LIVING REVIEWS IN RELATIVITY
, 2005
"... In three spacetime dimensions, general relativity drastically simplifies, becoming a “topological” theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body o ..."
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Cited by 137 (0 self)
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In three spacetime dimensions, general relativity drastically simplifies, becoming a “topological” theory with no propagating local degrees of freedom. Nevertheless, many of the difficult conceptual problems of quantizing gravity are still present. In this review, I summarize the rather large body of work that has gone towards quantizing (2+1)dimensional vacuum gravity in the setting of a spatially closed universe.
The WeilPetersson metric and volumes of 3dimensional hyperbolic convex cores
 J. Amer. Math. Soc
, 2001
"... Recent insights into the combinatorial geometry of Teichmüller space have shed new light on fundamental questions in hyperbolic geometry in 2 and 3 dimensions. Paradoxically, a coarse perspective on Teichmüller space appears to refine the analogy of Teichmüller geometry with the internal geometry of ..."
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Cited by 74 (10 self)
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Recent insights into the combinatorial geometry of Teichmüller space have shed new light on fundamental questions in hyperbolic geometry in 2 and 3 dimensions. Paradoxically, a coarse perspective on Teichmüller space appears to refine the analogy of Teichmüller geometry with the internal geometry of hyperbolic 3manifolds
The volume of hyperbolic alternating link complements. With an appendix by Ian Agol and Dylan Thurston
 Zbl 1041.57002 MR 2018964
"... A major goal of knot theory is to relate the geometric structure of a knot complement to the knot’s more basic topological properties. In this paper, we will do this for hyperbolic alternating knots and links, by showing that the link’s most fundamental geometric invariant its volume can be estima ..."
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Cited by 60 (2 self)
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A major goal of knot theory is to relate the geometric structure of a knot complement to the knot’s more basic topological properties. In this paper, we will do this for hyperbolic alternating knots and links, by showing that the link’s most fundamental geometric invariant its volume can be estimated directly from its
Dehn filling in relatively hyperbolic groups
 ISRAEL JOURNAL OF MATHEMATICS
, 2007
"... We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we con ..."
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Cited by 46 (6 self)
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We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the relative hyperbolicity of G in many natural ways. Second, we construct two useful bicombings on this space. The first of these, preferred paths, is combinatorial in nature and allows us to define the second, a relatively hyperbolic version of a construction of Mineyev. As an application, we prove a grouptheoretic analog of the GromovThurston 2π Theorem in the context of relatively hyperbolic groups.
On the density of geometrically finite Kleinian groups
, 2002
"... The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3manifolds. We prove that the conjecture obtains for each complete hyperbolic 3manifold with ..."
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Cited by 42 (10 self)
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The density conjecture of Bers, Sullivan and Thurston predicts that each complete hyperbolic 3manifold M with finitely generated fundamental group is an algebraic limit of geometrically finite hyperbolic 3manifolds. We prove that the conjecture obtains for each complete hyperbolic 3manifold with no cusps and incompressible ends.
Cusps Are Dense
, 1994
"... We show cusps are dense in Bers' boundary for Teichmüller space. The proof rests on an estimate for the algebraic effect of a unit quasiconformal deformation supported in the thin part of a hyperbolic Riemann surface. ..."
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Cited by 40 (2 self)
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We show cusps are dense in Bers' boundary for Teichmüller space. The proof rests on an estimate for the algebraic effect of a unit quasiconformal deformation supported in the thin part of a hyperbolic Riemann surface.
Discrete Surface Ricci Flow
 SUBMITTED TO IEEE TVCG
"... This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by userdefined Gaussian curvatures. Furthermore, the target metrics are conform ..."
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Cited by 40 (22 self)
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This work introduces a unified framework for discrete surface Ricci flow algorithms, including spherical, Euclidean, and hyperbolic Ricci flows, which can design Riemannian metrics on surfaces with arbitrary topologies by userdefined Gaussian curvatures. Furthermore, the target metrics are conformal (anglepreserving) to the original metrics. A Ricci flow conformally deforms the Riemannian metric on a surface according to its induced curvature, such that the curvature evolves like a heat diffusion process. Eventually, the curvature becomes the user defined curvature. Discrete Ricci flow algorithms are based on a variational framework. Given a mesh, all possible metrics form a linear space, and all possible curvatures form a convex polytope. The Ricci energy is defined on the metric space, which reaches its minimum at the desired metric. The Ricci flow is the negative gradient flow of the Ricci energy. Furthermore, the Ricci energy can be optimized using Newton’s method more efficiently. Discrete Ricci flow algorithms are rigorous and efficient. Our experimental results demonstrate the efficiency, accuracy and flexibility of the algorithms. They have the potential for a wide range of applications in graphics, geometric modeling, and medical imaging. We demonstrate their practical values by global surface parameterizations.
Greedy routing with guaranteed delivery using ricci flows
 In Proc. of the 8th International Symposium on Information Processing in Sensor Networks (IPSN’09
, 2009
"... Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In parti ..."
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Cited by 39 (17 self)
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Greedy forwarding with geographical locations in a wireless sensor network may fail at a local minimum. In this paper we propose to use conformal mapping to compute a new embedding of the sensor nodes in the plane such that greedy forwarding with the virtual coordinates guarantees delivery. In particular, we extract a planar triangulation of the sensor network with nontriangular faces as holes, by either using the nodes ’ location or using a landmarkbased scheme without node location. The conformal map is computed with Ricci flow such that all the nontriangular faces are mapped to perfect circles. Thus greedy forwarding will never get stuck at an intermediate node. The computation of the conformal map and the virtual coordinates is performed at a preprocessing phase and can be implemented by local gossipstyle computation. The method applies to both unit disk graph models and quasiunit disk graph models. Simulation results are presented for these scenarios.