The Geometry Of Discrete Groups

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The Geometry of Discrete Groups

Author : Alan F. Beardon
Publisher : Springer Science & Business Media
ISBN 13 : 1461211468
Total Pages : 350 pages
Book Rating : 4.4/5 (612 download)

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Book Synopsis The Geometry of Discrete Groups by : Alan F. Beardon

Download or read book The Geometry of Discrete Groups written by Alan F. Beardon and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 350 pages. Available in PDF, EPUB and Kindle. Book excerpt: This text is intended to serve as an introduction to the geometry of the action of discrete groups of Mobius transformations. The subject matter has now been studied with changing points of emphasis for over a hundred years, the most recent developments being connected with the theory of 3-manifolds: see, for example, the papers of Poincare [77] and Thurston [101]. About 1940, the now well-known (but virtually unobtainable) Fenchel-Nielsen manuscript appeared. Sadly, the manuscript never appeared in print, and this more modest text attempts to display at least some of the beautiful geo metrical ideas to be found in that manuscript, as well as some more recent material. The text has been written with the conviction that geometrical explana tions are essential for a full understanding of the material and that however simple a matrix proof might seem, a geometric proof is almost certainly more profitable. Further, wherever possible, results should be stated in a form that is invariant under conjugation, thus making the intrinsic nature of the result more apparent. Despite the fact that the subject matter is concerned with groups of isometries of hyperbolic geometry, many publications rely on Euclidean estimates and geometry. However, the recent developments have again emphasized the need for hyperbolic geometry, and I have included a comprehensive chapter on analytical (not axiomatic) hyperbolic geometry. It is hoped that this chapter will serve as a "dictionary" offormulae in plane hyperbolic geometry and as such will be of interest and use in its own right.

Conformal Geometry of Discrete Groups and Manifolds

Author : Boris N. Apanasov
Publisher : Walter de Gruyter
ISBN 13 : 3110808056
Total Pages : 541 pages
Book Rating : 4.1/5 (18 download)

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Book Synopsis Conformal Geometry of Discrete Groups and Manifolds by : Boris N. Apanasov

Download or read book Conformal Geometry of Discrete Groups and Manifolds written by Boris N. Apanasov and published by Walter de Gruyter. This book was released on 2011-06-24 with total page 541 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of the Expositions is to present new and important developments in pure and applied mathematics. Well established in the community over more than two decades, the series offers a large library of mathematical works, including several important classics. The volumes supply thorough and detailed expositions of the methods and ideas essential to the topics in question. In addition, they convey their relationships to other parts of mathematics. The series is addressed to advanced readers interested in a thorough study of the subject. Editorial Board Lev Birbrair, Universidade Federal do Ceará, Fortaleza, Brasil Walter D. Neumann, Columbia University, New York, USA Markus J. Pflaum, University of Colorado, Boulder, USA Dierk Schleicher, Jacobs University, Bremen, Germany Katrin Wendland, University of Freiburg, Germany Honorary Editor Victor P. Maslov, Russian Academy of Sciences, Moscow, Russia Titles in planning include Yuri A. Bahturin, Identical Relations in Lie Algebras (2019) Yakov G. Berkovich, Lev G. Kazarin, and Emmanuel M. Zhmud', Characters of Finite Groups, Volume 2 (2019) Jorge Herbert Soares de Lira, Variational Problems for Hypersurfaces in Riemannian Manifolds (2019) Volker Mayer, Mariusz Urbański, and Anna Zdunik, Random and Conformal Dynamical Systems (2021) Ioannis Diamantis, Boštjan Gabrovšek, Sofia Lambropoulou, and Maciej Mroczkowski, Knot Theory of Lens Spaces (2021)

Hyperbolic Manifolds and Discrete Groups

Author : Michael Kapovich
Publisher : Springer Science & Business Media
ISBN 13 : 0817649131
Total Pages : 470 pages
Book Rating : 4.8/5 (176 download)

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Book Synopsis Hyperbolic Manifolds and Discrete Groups by : Michael Kapovich

Download or read book Hyperbolic Manifolds and Discrete Groups written by Michael Kapovich and published by Springer Science & Business Media. This book was released on 2009-08-04 with total page 470 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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.

Bounded Cohomology of Discrete Groups

Author : Roberto Frigerio
Publisher : American Mathematical Soc.
ISBN 13 : 1470441462
Total Pages : 193 pages
Book Rating : 4.4/5 (74 download)

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Book Synopsis Bounded Cohomology of Discrete Groups by : Roberto Frigerio

Download or read book Bounded Cohomology of Discrete Groups written by Roberto Frigerio and published by American Mathematical Soc.. This book was released on 2017-11-21 with total page 193 pages. Available in PDF, EPUB and Kindle. Book excerpt: The theory of bounded cohomology, introduced by Gromov in the late 1980s, has had powerful applications in geometric group theory and the geometry and topology of manifolds, and has been the topic of active research continuing to this day. This monograph provides a unified, self-contained introduction to the theory and its applications, making it accessible to a student who has completed a first course in algebraic topology and manifold theory. The book can be used as a source for research projects for master's students, as a thorough introduction to the field for graduate students, and as a valuable landmark text for researchers, providing both the details of the theory of bounded cohomology and links of the theory to other closely related areas. The first part of the book is devoted to settling the fundamental definitions of the theory, and to proving some of the (by now classical) results on low-dimensional bounded cohomology and on bounded cohomology of topological spaces. The second part describes applications of the theory to the study of the simplicial volume of manifolds, to the classification of circle actions, to the analysis of maximal representations of surface groups, and to the study of flat vector bundles with a particular emphasis on the possible use of bounded cohomology in relation with the Chern conjecture. Each chapter ends with a discussion of further reading that puts the presented results in a broader context.

Discrete Groups, Expanding Graphs and Invariant Measures

Author : Alex Lubotzky
Publisher : Springer Science & Business Media
ISBN 13 : 3034603320
Total Pages : 201 pages
Book Rating : 4.0/5 (346 download)

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Book Synopsis Discrete Groups, Expanding Graphs and Invariant Measures by : Alex Lubotzky

Download or read book Discrete Groups, Expanding Graphs and Invariant Measures written by Alex Lubotzky and published by Springer Science & Business Media. This book was released on 2010-02-17 with total page 201 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the last ?fteen years two seemingly unrelated problems, one in computer science and the other in measure theory, were solved by amazingly similar techniques from representation theory and from analytic number theory. One problem is the - plicit construction of expanding graphs («expanders»). These are highly connected sparse graphs whose existence can be easily demonstrated but whose explicit c- struction turns out to be a dif?cult task. Since expanders serve as basic building blocks for various distributed networks, an explicit construction is highly des- able. The other problem is one posed by Ruziewicz about seventy years ago and studied by Banach [Ba]. It asks whether the Lebesgue measure is the only ?nitely additive measure of total measure one, de?ned on the Lebesgue subsets of the n-dimensional sphere and invariant under all rotations. The two problems seem, at ?rst glance, totally unrelated. It is therefore so- what surprising that both problems were solved using similar methods: initially, Kazhdan’s property (T) from representation theory of semi-simple Lie groups was applied in both cases to achieve partial results, and later on, both problems were solved using the (proved) Ramanujan conjecture from the theory of automorphic forms. The fact that representation theory and automorphic forms have anything to do with these problems is a surprise and a hint as well that the two questions are strongly related.

Geometry, Analysis and Topology of Discrete Groups

Author : Lizhen Ji
Publisher :
ISBN 13 :
Total Pages : 504 pages
Book Rating : 4.3/5 (91 download)

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Book Synopsis Geometry, Analysis and Topology of Discrete Groups by : Lizhen Ji

Download or read book Geometry, Analysis and Topology of Discrete Groups written by Lizhen Ji and published by . This book was released on 2008 with total page 504 pages. Available in PDF, EPUB and Kindle. Book excerpt: Presents 15 papers treating discrete groups as they occur in areas such as algebra, analysis, geometry, number theory and topology. This work helps graduate students and researchers to understand the structures and applications of discrete subgroups of Lie groups and locally symmetric spaces.

The Ergodic Theory of Discrete Groups

Author : Peter J. Nicholls
Publisher : Cambridge University Press
ISBN 13 : 0521376742
Total Pages : 237 pages
Book Rating : 4.5/5 (213 download)

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Book Synopsis The Ergodic Theory of Discrete Groups by : Peter J. Nicholls

Download or read book The Ergodic Theory of Discrete Groups written by Peter J. Nicholls and published by Cambridge University Press. This book was released on 1989-08-17 with total page 237 pages. Available in PDF, EPUB and Kindle. Book excerpt: The interaction between ergodic theory and discrete groups has a long history and much work was done in this area by Hedlund, Hopf and Myrberg in the 1930s. There has been a great resurgence of interest in the field, due in large measure to the pioneering work of Dennis Sullivan. Tools have been developed and applied with outstanding success to many deep problems. The ergodic theory of discrete groups has become a substantial field of mathematical research in its own right, and it is the aim of this book to provide a rigorous introduction from first principles to some of the major aspects of the theory. The particular focus of the book is on the remarkable measure supported on the limit set of a discrete group that was first developed by S. J. Patterson for Fuchsian groups, and later extended and refined by Sullivan.

Discrete Groups in Geometry and Analysis

Author : Howe
Publisher : Springer Science & Business Media
ISBN 13 : 1489966641
Total Pages : 223 pages
Book Rating : 4.4/5 (899 download)

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Book Synopsis Discrete Groups in Geometry and Analysis by : Howe

Download or read book Discrete Groups in Geometry and Analysis written by Howe and published by Springer Science & Business Media. This book was released on 2013-11-22 with total page 223 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Geometry of Crystallographic Groups

Author : Andrzej Szczepański
Publisher : World Scientific
ISBN 13 : 9814412252
Total Pages : 208 pages
Book Rating : 4.8/5 (144 download)

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Book Synopsis Geometry of Crystallographic Groups by : Andrzej Szczepański

Download or read book Geometry of Crystallographic Groups written by Andrzej Szczepański and published by World Scientific. This book was released on 2012 with total page 208 pages. Available in PDF, EPUB and Kindle. Book excerpt: Crystallographic groups are groups which act in a nice way and via isometries on some n-dimensional Euclidean space. This book gives an example of the torsion free crystallographic group with a trivial center and a trivial outer automorphism group.

Foundations of Hyperbolic Manifolds

Author : John Ratcliffe
Publisher : Springer Science & Business Media
ISBN 13 : 1475740131
Total Pages : 761 pages
Book Rating : 4.4/5 (757 download)

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Book Synopsis Foundations of Hyperbolic Manifolds by : John Ratcliffe

Download or read book Foundations of Hyperbolic Manifolds written by John Ratcliffe and published by Springer Science & Business Media. This book was released on 2013-03-09 with total page 761 pages. Available in PDF, EPUB and Kindle. Book excerpt: 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. Particular emphasis has been placed on readability and completeness of ar gument. The treatment of the material is for the most part elementary and self-contained. 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, consisting of Chap ters 1-7, is concerned with hyperbolic geometry and basic properties of discrete groups of isometries of hyperbolic space. The main results are the existence theorem for discrete reflection groups, the Bieberbach theorems, and Selberg's lemma. The second part, consisting of Chapters 8-12, is de voted to the theory of hyperbolic manifolds. The main results are Mostow's rigidity theorem and the determination of the structure of geometrically finite hyperbolic manifolds. The third part, consisting of Chapter 13, in tegrates the first two parts in a development of the theory of hyperbolic orbifolds. The main results are the construction of the universal orbifold covering space and Poincare's fundamental polyhedron theorem.

Geometries and Groups

Author : Viacheslav V. Nikulin
Publisher : Springer Science & Business Media
ISBN 13 : 3642615708
Total Pages : 262 pages
Book Rating : 4.6/5 (426 download)

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Book Synopsis Geometries and Groups by : Viacheslav V. Nikulin

Download or read book Geometries and Groups written by Viacheslav V. Nikulin and published by Springer Science & Business Media. This book was released on 2012-12-06 with total page 262 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the theory of geometries which are locally Euclidean, in the sense that in small regions they are identical to the geometry of the Euclidean plane or Euclidean 3-space. Starting from the simplest examples, we proceed to develop a general theory of such geometries, based on their relation with discrete groups of motions of the Euclidean plane or 3-space; we also consider the relation between discrete groups of motions and crystallography. The description of locally Euclidean geometries of one type shows that these geometries are themselves naturally represented as the points of a new geometry. The systematic study of this new geometry leads us to 2-dimensional Lobachevsky geometry (also called non-Euclidean or hyperbolic geometry) which, following the logic of our study, is constructed starting from the properties of its group of motions. Thus in this book we would like to introduce the reader to a theory of geometries which are different from the usual Euclidean geometry of the plane and 3-space, in terms of examples which are accessible to a concrete and intuitive study. The basic method of study is the use of groups of motions, both discrete groups and the groups of motions of geometries. The book does not presuppose on the part of the reader any preliminary knowledge outside the limits of a school geometry course.

Discrete Subgroups of Semisimple Lie Groups

Author : Gregori A. Margulis
Publisher : Springer Science & Business Media
ISBN 13 : 9783540121794
Total Pages : 408 pages
Book Rating : 4.1/5 (217 download)

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Book Synopsis Discrete Subgroups of Semisimple Lie Groups by : Gregori A. Margulis

Download or read book Discrete Subgroups of Semisimple Lie Groups written by Gregori A. Margulis and published by Springer Science & Business Media. This book was released on 1991-02-15 with total page 408 pages. Available in PDF, EPUB and Kindle. Book excerpt: Discrete subgroups have played a central role throughout the development of numerous mathematical disciplines. Discontinuous group actions and the study of fundamental regions are of utmost importance to modern geometry. Flows and dynamical systems on homogeneous spaces have found a wide range of applications, and of course number theory without discrete groups is unthinkable. This book, written by a master of the subject, is primarily devoted to discrete subgroups of finite covolume in semi-simple Lie groups. Since the notion of "Lie group" is sufficiently general, the author not only proves results in the classical geometry setting, but also obtains theorems of an algebraic nature, e.g. classification results on abstract homomorphisms of semi-simple algebraic groups over global fields. The treatise of course contains a presentation of the author's fundamental rigidity and arithmeticity theorems. The work in this monograph requires the language and basic results from fields such as algebraic groups, ergodic theory, the theory of unitary representatons, and the theory of amenable groups. The author develops the necessary material from these subjects; so that, while the book is of obvious importance for researchers working in related areas, it is essentially self-contained and therefore is also of great interest for advanced students.

Discrete Groups and Geometry

Author : William J. Harvey
Publisher : Cambridge University Press
ISBN 13 : 0521429323
Total Pages : 260 pages
Book Rating : 4.5/5 (214 download)

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Book Synopsis Discrete Groups and Geometry by : William J. Harvey

Download or read book Discrete Groups and Geometry written by William J. Harvey and published by Cambridge University Press. This book was released on 1992-07-30 with total page 260 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book constitutes the proceedings of a conference held at the University of Birmingham to mark the retirement of Professor A. M. Macbeath. The papers represent up-to-date work on a broad spectrum of topics in the theory of discrete group actions, ranging from presentations of finite groups through the detailed study of Fuchsian and crystallographic groups, to applications of group actions in low dimensional topology, complex analysis, algebraic geometry and number theory. For those wishing to pursue research in these areas, this volume offers a valuable summary of contemporary thought and a source of fresh geometric insights.

Discrete Groups in Geometry and Analysis

Author : R. Howe
Publisher :
ISBN 13 : 9781489966650
Total Pages : 228 pages
Book Rating : 4.9/5 (666 download)

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Book Synopsis Discrete Groups in Geometry and Analysis by : R. Howe

Download or read book Discrete Groups in Geometry and Analysis written by R. Howe and published by . This book was released on 2014-09-01 with total page 228 pages. Available in PDF, EPUB and Kindle. Book excerpt:

Geometry and Discrete Mathematics

Author : Benjamin Fine
Publisher : Walter de Gruyter GmbH & Co KG
ISBN 13 : 3110740788
Total Pages : 364 pages
Book Rating : 4.1/5 (17 download)

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Book Synopsis Geometry and Discrete Mathematics by : Benjamin Fine

Download or read book Geometry and Discrete Mathematics written by Benjamin Fine and published by Walter de Gruyter GmbH & Co KG. This book was released on 2022-08-22 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: Fundamentals of mathematics are presented in the two-volume set in an exciting and pedagogically sound way. The present volume examines the most important basic results in geometry and discrete mathematics, along with their proofs, and also their history. New: A chapter on discrete Morse theory and still more graph theory for solving further classical problems as the Travelling Salesman and Postman problem.

Discrete Groups and Geometric Structures

Author : Karel Dekimpe
Publisher : American Mathematical Soc.
ISBN 13 : 0821846477
Total Pages : 146 pages
Book Rating : 4.8/5 (218 download)

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Book Synopsis Discrete Groups and Geometric Structures by : Karel Dekimpe

Download or read book Discrete Groups and Geometric Structures written by Karel Dekimpe and published by American Mathematical Soc.. This book was released on 2009-11-12 with total page 146 pages. Available in PDF, EPUB and Kindle. Book excerpt: This volume reports on research related to Discrete Groups and Geometric Structures, as presented during the International Workshop held May 26-30, 2008, in Kortrijk, Belgium. Readers will benefit from impressive survey papers by John R. Parker on methods to construct and study lattices in complex hyperbolic space and by Ursula Hamenstadt on properties of group actions with a rank-one element on proper $\mathrm{CAT}(0)$-spaces. This volume also contains research papers in the area of group actions and geometric structures, including work on loops on a twice punctured torus, the simplicial volume of products and fiber bundles, the homology of Hantzsche-Wendt groups, rigidity of real Bott towers, circles in groups of smooth circle homeomorphisms, and groups generated by spine reflections admitting crooked fundamental domains.

The Geometry and Topology of Coxeter Groups

Author : Michael Davis
Publisher : Princeton University Press
ISBN 13 : 0691131384
Total Pages : 601 pages
Book Rating : 4.6/5 (911 download)

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Book Synopsis The Geometry and Topology of Coxeter Groups by : Michael Davis

Download or read book The Geometry and Topology of Coxeter Groups written by Michael Davis and published by Princeton University Press. This book was released on 2008 with total page 601 pages. Available in PDF, EPUB and Kindle. Book excerpt: The Geometry and Topology of Coxeter Groups is a comprehensive and authoritative treatment of Coxeter groups from the viewpoint of geometric group theory. Groups generated by reflections are ubiquitous in mathematics, and there are classical examples of reflection groups in spherical, Euclidean, and hyperbolic geometry. Any Coxeter group can be realized as a group generated by reflection on a certain contractible cell complex, and this complex is the principal subject of this book. The book explains a theorem of Moussong that demonstrates that a polyhedral metric on this cell complex is nonpositively curved, meaning that Coxeter groups are "CAT(0) groups." The book describes the reflection group trick, one of the most potent sources of examples of aspherical manifolds. And the book discusses many important topics in geometric group theory and topology, including Hopf's theory of ends; contractible manifolds and homology spheres; the Poincaré Conjecture; and Gromov's theory of CAT(0) spaces and groups. Finally, the book examines connections between Coxeter groups and some of topology's most famous open problems concerning aspherical manifolds, such as the Euler Characteristic Conjecture and the Borel and Singer conjectures.