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Foundations of Hyperbolic Manifolds

01 Jan 1994-
TL;DR: In this paper, an exposition of the theoretical foundations of hyperbolic manifolds is presented, which is intended to be used both as a textbook and as a reference for algebra and topology courses.
Abstract: This book is an exposition of the theoretical foundations of hyperbolic manifolds. It is intended to be used both as a textbook and as a reference. The reader is assumed to have a basic knowledge of algebra and topology at the first year graduate level of an American university. The book is divided into three parts. The first part, Chapters 1-7, is concerned with hyperbolic geometry and discrete groups. The second part, Chapters 8-12, is devoted to the theory of hyperbolic manifolds. The third part, Chapter 13, integrates the first two parts in a development of the theory of hyperbolic orbifolds. There are over 500 exercises in this book and more than 180 illustrations.
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Book
17 Jan 1997
TL;DR: In this article, the Structure of Discrete Groups (SDSG) is defined as a set of discrete groups that can be represented by a geometric manifold, and the structure of the manifold is discussed.
Abstract: Preface Reader's Advisory Ch. 1. What Is a Manifold? 3 Ch. 2. Hyperbolic Geometry and Its Friends 43 Ch. 3. Geometric Manifolds 109 Ch. 4. The Structure of Discrete Groups 209 Glossary 289 Bibliography 295 Index 301

1,834 citations

Journal ArticleDOI
TL;DR: It is shown that targeted transport processes without global topology knowledge are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure.
Abstract: We develop a geometric framework to study the structure and function of complex networks. We assume that hyperbolic geometry underlies these networks, and we show that with this assumption, heterogeneous degree distributions and strong clustering in complex networks emerge naturally as simple reflections of the negative curvature and metric property of the underlying hyperbolic geometry. Conversely, we show that if a network has some metric structure, and if the network degree distribution is heterogeneous, then the network has an effective hyperbolic geometry underneath. We then establish a mapping between our geometric framework and statistical mechanics of complex networks. This mapping interprets edges in a network as noninteracting fermions whose energies are hyperbolic distances between nodes, while the auxiliary fields coupled to edges are linear functions of these energies or distances. The geometric network ensemble subsumes the standard configuration model and classical random graphs as two limiting cases with degenerate geometric structures. Finally, we show that targeted transport processes without global topology knowledge, made possible by our geometric framework, are maximally efficient, according to all efficiency measures, in networks with strongest heterogeneity and clustering, and that this efficiency is remarkably robust with respect to even catastrophic disturbances and damages to the network structure.

1,002 citations


Cites background from "Foundations of Hyperbolic Manifolds..."

  • ...There are only three types of isotropic spaces: Eu- clidean (flat), spherical (positively curved), and hyperbolic (negatively curved)....

    [...]

Book
01 Jan 1993
TL;DR: In this article, the authors propose a discrete groups of motions of Spaces of Constant Curvature for the problem of space-of-constant-curvature geometry.
Abstract: I. Geometry of Spaces of Constant Curvature.- II. Discrete Groups of Motions of Spaces of Constant Curvature.- Author Index.

383 citations

MonographDOI
TL;DR: In this paper, the authors summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.
Abstract: We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

336 citations

Journal ArticleDOI
TL;DR: In this article, the authors introduce spin foam models for non-perturbative quantum gravity, an approach that lies at the point of convergence of many different research areas, including loop quantum gravity and topological quantum field theories.
Abstract: This is an introduction to spin foam models for non-perturbative quantum gravity, an approach that lies at the point of convergence of many different research areas, including loop quantum gravity, topological quantum field theories, path integral quantum gravity, lattice field theory, matrix models, category theory and statistical mechanics. We describe the general formalism and ideas of spin foam models, the picture of quantum geometry emerging from them, and give a review of the results obtained so far, in both the Euclidean and Lorentzian cases. We focus in particular on the Barrett-Crane model for four-dimensional quantum gravity.

214 citations