Outer Circles

Author: A. Marden
Publisher: Cambridge University Press
ISBN: 1139463764
Format: PDF, Mobi
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We live in a three-dimensional space; what sort of space is it? Can we build it from simple geometric objects? The answers to such questions have been found in the last 30 years, and Outer Circles describes the basic mathematics needed for those answers as well as making clear the grand design of the subject of hyperbolic manifolds as a whole. The purpose of Outer Circles is to provide an account of the contemporary theory, accessible to those with minimal formal background in topology, hyperbolic geometry, and complex analysis. The text explains what is needed, and provides the expertise to use the primary tools to arrive at a thorough understanding of the big picture. This picture is further filled out by numerous exercises and expositions at the ends of the chapters and is complemented by a profusion of high quality illustrations. There is an extensive bibliography for further study.

The Arithmetic of Hyperbolic 3 Manifolds

Author: Colin Maclachlan
Publisher: Springer Science & Business Media
ISBN: 9780387983868
Format: PDF, ePub, Docs
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This text is based on graduate courses given by the authors on one of the most active areas of current research. It brings together much of the existing literature on arithmetic Kleinan groups in a clear and concise way, containing many examples and lots of problems.

Buildings Finite Geometries and Groups

Author: N.S. Narasimha Sastry
Publisher: Springer Science & Business Media
ISBN: 1461407095
Format: PDF, ePub, Docs
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This is the Proceedings of the ICM 2010 Satellite Conference on “Buildings, Finite Geometries and Groups” organized at the Indian Statistical Institute, Bangalore, during August 29 – 31, 2010. This is a collection of articles by some of the currently very active research workers in several areas related to finite simple groups, Chevalley groups and their generalizations: theory of buildings, finite incidence geometries, modular representations, Lie theory, etc. These articles reflect the current major trends in research in the geometric and combinatorial aspects of the study of these groups. The unique perspective the authors bring in their articles on the current developments and the major problems in their area is expected to be very useful to research mathematicians, graduate students and potential new entrants to these areas.

Hyperbolic Manifolds and Discrete Groups

Author: Michael Kapovich
Publisher: Springer Science & Business Media
ISBN: 9780817639044
Format: PDF, Kindle
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Hyperbolic Manifolds and Discrete Groups is at the crossroads of several branches of mathematics: hyperbolic geometry, discrete groups, 3-dimensional topology, geometric group theory, and complex analysis. The main focus throughout the text is on the "Big Monster," i.e., on Thurston’s hyperbolization theorem, which has not only completely changes the landscape of 3-dimensinal topology and Kleinian group theory but is one of the central results of 3-dimensional topology. The book is fairly self-contained, replete with beautiful illustrations, a rich set of examples of key concepts, numerous exercises, and an extensive bibliography and index. It should serve as an ideal graduate course/seminar text or as a comprehensive reference.

Homotopy Equivalences of 3 Manifolds and Deformation Theory of Kleinian Groups

Author: Richard Douglas Canary
Publisher: American Mathematical Soc.
ISBN: 0821835491
Format: PDF, ePub, Mobi
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This text investigates a natural question arising in the topological theory of $3$-manifolds, and applies the results to give new information about the deformation theory of hyperbolic $3$-manifolds. It is well known that some compact $3$-manifolds with boundary admit homotopy equivalences that are not homotopic to homeomorphisms. We investigate when the subgroup $\mathcal{R}(M)$ of outer automorphisms of $\pi_1(M)$ which are induced by homeomorphisms of a compact $3$-manifold $M$ has finite index in the group $\operatorname{Out}(\pi_1(M))$ of all outer automorphisms. This question is completely resolved for Haken $3$-manifolds.It is also resolved for many classes of reducible $3$-manifolds and $3$-manifolds with boundary patterns, including all pared $3$-manifolds. The components of the interior $\operatorname{GF}(\pi_1(M))$ of the space $\operatorname{AH}(\pi_1(M))$ of all (marked) hyperbolic $3$-manifolds homotopy equivalent to $M$ are enumerated by the marked homeomorphism types of manifolds homotopy equivalent to $M$, so one may apply the topological results above to study the topology of this deformation space.We show that $\operatorname{GF}(\pi_1(M))$ has finitely many components if and only if either $M$ has incompressible boundary, but no 'double trouble', or $M$ has compressible boundary and is 'small'. (A hyperbolizable $3$-manifold with incompressible boundary has double trouble if and only if there is a thickened torus component of its characteristic submanifold which intersects the boundary in at least two annuli). More generally, the deformation theory of hyperbolic structures on pared manifolds is analyzed. Some expository sections detail Johannson's formulation of the Jaco-Shalen-Johannson characteristic submanifold theory, the topology of pared $3$-manifolds, and the deformation theory of hyperbolic $3$-manifolds. An epilogue discusses related open problems and recent progress in the deformation theory of hyperbolic $3$-manifolds.

Hyperbolic Manifolds

Author: Albert Marden
Publisher: Cambridge University Press
ISBN: 1316432521
Format: PDF, Docs
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Over the past three decades there has been a total revolution in the classic branch of mathematics called 3-dimensional topology, namely the discovery that most solid 3-dimensional shapes are hyperbolic 3-manifolds. This book introduces and explains hyperbolic geometry and hyperbolic 3- and 2-dimensional manifolds in the first two chapters and then goes on to develop the subject. The author discusses the profound discoveries of the astonishing features of these 3-manifolds, helping the reader to understand them without going into long, detailed formal proofs. The book is heavily illustrated with pictures, mostly in color, that help explain the manifold properties described in the text. Each chapter ends with a set of exercises and explorations that both challenge the reader to prove assertions made in the text, and suggest further topics to explore that bring additional insight. There is an extensive index and bibliography.

Lectures on Quasiconformal Mappings

Author: Lars Valerian Ahlfors
Publisher: American Mathematical Soc.
ISBN: 0821836447
Format: PDF, Docs
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Lars Ahlfors's Lectures on Quasiconformal Mappings, based on a course he gave at Harvard University in the spring term of 1964, was first published in 1966 and was soon recognized as the classic it was shortly destined to become. These lectures develop the theory of quasiconformal mappings from scratch, give a self-contained treatment of the Beltrami equation, and cover the basic properties of Teichmuller spaces, including the Bers embedding and the Teichmuller curve. It is remarkable how Ahlfors goes straight to the heart of the matter, presenting major results with a minimum set of prerequisites. Many graduate students and other mathematicians have learned the foundations of the theories of quasiconformal mappings and Teichmuller spaces from these lecture notes. This edition includes three new chapters. The first, written by Earle and Kra, describes further developments in the theory of Teichmuller spaces and provides many references to the vast literature on Teichmuller spaces and quasiconformal mappings. The second, by Shishikura, describes how quasiconformal mappings have revitalized the subject of complex dynamics. The third, by Hubbard, illustrates the role of these mappings in Thurston's theory of hyperbolic structures on 3-manifolds. Together, these three new chapters exhibit the continuing vitality and importance of the theory of quasiconformal mappings.

Two generator Discrete Subgroups of PSL 2 R

Author: Jane Gilman
Publisher: American Mathematical Soc.
ISBN: 9780821862841
Format: PDF, ePub
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The discreteness problem is the problem of determining whether or not a two-generator subgroup of $PSL(2, R)$ is discrete. Historically, papers on this old and subtle problem have been known for their errors and omissions. This book presents the first complete geometric solution to the discreteness problem by building upon cases previously presented by Gilman and Maskit and by developing a theory of triangle group shinglings/tilings of the hyperbolic plane and a theory explaining why the solution must take the form of an algorithm. This work is a thoroughly readable exposition that captures the beauty of the interplay between the algebra and the geometry of the solution.