Results 1  10
of
74
Embeddings Of Gromov Hyperbolic Spaces
 Geom. Funct. Anal
"... . It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasiisometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at m ..."
Abstract

Cited by 54 (6 self)
 Add to MetaCart
. It is shown that a Gromov hyperbolic geodesic metric space X with bounded growth at some scale is roughly quasiisometric to a convex subset of hyperbolic space. If one is allowed to rescale the metric of X by some positive constant, then there is an embedding where distances are distorted by at most an additive constant. Another embedding theorem states that any ffi hyperbolic metric space embeds isometrically into a complete geodesic ffi hyperbolic space. The relation of a Gromov hyperbolic space to its boundary is further investigated. One of the applications is a characterization of the hyperbolic plane up to rough quasiisometries. 1. Introduction The study of Gromov hyperbolic spaces has been largely motivated and dominated by questions about Gromov hyperbolic groups. This paper studies the geometry of Gromov hyperbolic spaces without reference to any group or group action. One of our main theorems is 1.1. Embedding Theorem. Let X be a Gromov hyperbolic geodesic metric spa...
On groups generated by two positive multitwists: Teichmüller curves and Lehmer’s number
, 2008
"... Following Thurston, we study subgroups of the mapping class group generated by two positive multitwists. We classify the configurations of curves for which the corresponding groups exhibit certain exceptional behaviors. We also identify a pseudoAnosov automorphism whose dilatation is Lehmer’s numb ..."
Abstract

Cited by 23 (2 self)
 Add to MetaCart
Following Thurston, we study subgroups of the mapping class group generated by two positive multitwists. We classify the configurations of curves for which the corresponding groups exhibit certain exceptional behaviors. We also identify a pseudoAnosov automorphism whose dilatation is Lehmer’s number, and show that this is minimal for the groups under consideration. Connections with Coxeter groups, billiards, and knot theory are also observed.
The Volume Spectrum Of Hyperbolic 4Manifolds
 Experiment. Math
"... . In this paper, we construct examples of complete hyperbolic 4manifolds of smallest volume by gluing together the sides of a regular ideal 24cell in hyperbolic 4space. We also show that the volume spectrum of hyperbolic 4manifolds is the set of all positive integral multiples of 4 2 =3. 1. ..."
Abstract

Cited by 22 (5 self)
 Add to MetaCart
(Show Context)
. In this paper, we construct examples of complete hyperbolic 4manifolds of smallest volume by gluing together the sides of a regular ideal 24cell in hyperbolic 4space. We also show that the volume spectrum of hyperbolic 4manifolds is the set of all positive integral multiples of 4 2 =3. 1. Introduction A hyperbolic manifold is a Riemannian manifold of constant sectional curvature \Gamma1. The set of all volumes of complete hyperbolic nmanifolds of finite volume is called the volume spectrum of hyperbolic nmanifolds. It has been known for over a hundred years that the volume spectrum of hyperbolic 2manifolds is the set of all positive integral multiples of 2. In contrast to dimension two, Jrgensen and Thurston [10] have shown that the volume spectrum of hyperbolic 3manifolds is a closed, nondiscrete, wellordered subset of the positive real numbers of order type ! ! . In particular, there is a smallest volume of a complete hyperbolic 3manifold. The smallest volume of...
On the rigidity of conformally compact Einstein manifolds
 Int. Math. Res. Not
"... Abstract. In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is ba ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
(Show Context)
Abstract. In this paper we prove that a conformally compact Einstein manifold with the round sphere as its conformal infinity has to be the hyperbolic space. We do not assume the manifolds to be spin, but our approach relies on the positive mass theorem for asymptotic flat manifolds. The proof is based on understanding of positive eigenfunctions and compactifications obtained by positive eigenfunctions. In this paper we study the rigidity problem for conformally compact Einstein manifolds with the round sphere as their conformal infinity. Quite recently there has been a great deal of interest in both physics and mathematics community in the socalled AntideSitter/Conformal Field Theory (in short AdS/CFT) correspondence.
Uniformizing Dessins And Belyi Maps Via Circle Packing
, 1997
"... Grothendieck's theory of Dessins d'Enfants involves combinatorially determined affine, reflective, and conformal structures on compact surfaces. In this paper the authors establish the first general method for uniformizing these dessin surfaces and for approximating their associated Be ..."
Abstract

Cited by 17 (6 self)
 Add to MetaCart
Grothendieck's theory of Dessins d'Enfants involves combinatorially determined affine, reflective, and conformal structures on compact surfaces. In this paper the authors establish the first general method for uniformizing these dessin surfaces and for approximating their associated Belyi meromorphic functions. The paper begins by developing a discrete theory of dessins based on circle packing. This theory is surprisingly faithful, even at its coarsest stages, to the geometry of the classical theory, and it displays some new sources of richness; in particular, algrebraic number fields enter the theory in a new way. The paper goes on to show that the discrete dessin structures converge to their classical counterparts under a hexagonal refinement scheme. In addition, since the discrete objects are computable, circle packing provides opportunities both for routine experimentation and for large scale explicit computation. A range of examples up to genus 4 is given in the pape...
Canonical Wick rotations in 3dimensional gravity
, 2008
"... We develop a canonical Wick rotationrescaling theory in 3dimensional gravity. This includes (a) A simultaneous classification: this shows how generic maximal globally hyperbolic spacetimes of arbitrary constant curvature, which admit a complete Cauchy surface, as well as complex projective structu ..."
Abstract

Cited by 15 (7 self)
 Add to MetaCart
We develop a canonical Wick rotationrescaling theory in 3dimensional gravity. This includes (a) A simultaneous classification: this shows how generic maximal globally hyperbolic spacetimes of arbitrary constant curvature, which admit a complete Cauchy surface, as well as complex projective structures on arbitrary surfaces, are all different materializations of “more fundamental ” encoding structures. (b) Canonical geometric correlations: this shows how spacetimes of different curvature, that share a same encoding structure, are related to each other by canonical rescalings, and how they can be transformed by canonical Wick rotations in hyperbolic 3manifolds, that carry the appropriate asymptotic projective structure. Both Wick rotations and rescalings act along the canonical cosmological time and have universal rescaling functions. These correlations are functorial with respect to isomorphisms of the respective geometric categories. In particular, the theory applies to spacetimes with compact Cauchy surfaces. By Mess classification, for every fixed genus g ≥ 2 of a Cauchy surface S, and
TRIANGULATED RIEMANN SURFACES WITH BOUNDARY AND THE WEILPETERSSON POISSON STRUCTURE
, 2006
"... Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at ∂S perpendicularly are coordinates on the Teichmüller space T (S). We compute the WeilPetersson Poisson structure on T (S) in this system of coordinates and we prov ..."
Abstract

Cited by 10 (2 self)
 Add to MetaCart
(Show Context)
Given a Riemann surface with boundary S, the lengths of a maximal system of disjoint simple geodesic arcs on S that start and end at ∂S perpendicularly are coordinates on the Teichmüller space T (S). We compute the WeilPetersson Poisson structure on T (S) in this system of coordinates and we prove that it limits pointwise to the piecewiselinear Poisson structure defined by Kontsevich on the arc complex of S. As a byproduct of the proof, we obtain a formula for the firstorder variation of the distance between two closed geodesic under FenchelNielsen deformation.
Globally hyperbolic flat spacetimes
 J. Geom. Phys
"... Abstract. We consider (flat) Cauchycomplete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes model spacetimes including four ..."
Abstract

Cited by 9 (3 self)
 Add to MetaCart
(Show Context)
Abstract. We consider (flat) Cauchycomplete GH spacetimes, i.e., globally hyperbolic flat lorentzian manifolds admitting some Cauchy hypersurface on which the ambient lorentzian metric restricts as a complete riemannian metric. We define a family of such spacetimes model spacetimes including four subfamilies: translation spacetimes, Misner spacetimes, unipotent spacetimes, and Cauchyhyperbolic spacetimes (the last family undoubtfull the most interesting one is a generalization of standart spacetimes defined by G. Mess). We prove that, up to finite coverings and (twisted) products by euclidean linear spaces, any Cauchycomplete GH spacetime can be isometrically embedded in a model spacetime, or in a twisted product of a Cauchyhyperbolic spacetime by flat euclidean torus. We obtain as a corollary the classification of maximal GH spacetimes admitting closed Cauchy hypersurfaces. We also establish the existence of CMC foliations on every model spacetime. 1.
On the (non)coincidence of MilnorThurston homology theory with singular homology theory
 Pacific J. Math
, 1998
"... The paper investigates a homology theory based on the ideas of Milnor and Thurston that by considering measures on the set of all singular simplices one should get alternate possibilities for describing the cycles of classical homology theory. It suggests slight changes to Milnor’s and Thurston’s or ..."
Abstract

Cited by 9 (0 self)
 Add to MetaCart
(Show Context)
The paper investigates a homology theory based on the ideas of Milnor and Thurston that by considering measures on the set of all singular simplices one should get alternate possibilities for describing the cycles of classical homology theory. It suggests slight changes to Milnor’s and Thurston’s original definitions (giving differences for wild topological spaces only) which ensure that their homology theory is welldefined on all topological spaces. It further proves that MilnorThurston homology theory gives the same homology groups as the singular homology theory with real coefficients for all triangulable spaces. An example showing that the coincidence between these both homology theories does not hold for all topological spaces is also included. The idea of the MilnorThurston homology is, roughly speaking, that the usual homology groups with real coefficients can be defined by considering measures on the set of all singular simplices when using appropriate definitions, and hence that infinite chains can be defined, which is of some use in certain proofs because of the additional possibilities that this homology theory provides in representing the cycles. Since those measures which are concentrated on a finite number of singular simplices give the same behavior as the corresponding formal sum of these singular simplices in the singular homology theory, the definition of chains by measures as explained in the first sentence can be regarded as a generalisation of the usual chains. The possibility that such a homology theory can by used in the proofs of the Mostow and Gromov Theorems for hyperbolic manifolds was stated in Thurston’s famous preprint [Th] (pp. 6.66.7). Thurston there referred this idea to a joint work with Milnor and to a forthcoming paper of [MTh], which, however never appeared ([MR], 19771994). The use of this homology theory can be avoided in the proof of the Mostow Theorem, but its use is essential in a proof of the fact the threedimensional closed oriented hyperbolic manifolds with the same volume are isometric, if they can be mapped onto each other by a degreeone mapping. This is the theorem we are calling “Gromov Theorem”, since Thurston referred this statement
The E_tConstruction for Lattices, Spheres and Polytopes
"... We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction ..."
Abstract

Cited by 9 (7 self)
 Add to MetaCart
(Show Context)
We describe and analyze a new construction that produces new Eulerian lattices from old ones. It specializes to a construction that produces new strongly regular cellular spheres (whose face lattices are Eulerian). The construction