Groups And Geometries

Author : Viacheslav V. Nikulin,Igor R. Shafarevich
Publisher : Springer Science & Business Media
Page : 262 pages
File Size : 54,7 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9783642615702

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Geometries and Groups by Viacheslav V. Nikulin,Igor R. Shafarevich Pdf

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.

Author : Roger C. Lyndon
Publisher : Cambridge University Press
Page : 231 pages
File Size : 50,8 Mb
Release : 1985-03-14
Category : Mathematics
ISBN : 9780521316941

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Groups and Geometry by Roger C. Lyndon Pdf

This 1985 book is an introduction to certain central ideas in group theory and geometry. Professor Lyndon emphasises and exploits the well-known connections between the two subjects and leads the reader to the frontiers of current research at the time of publication.

Author : Vaughn Climenhaga,Anatole Katok
Publisher : American Mathematical Soc.
Page : 420 pages
File Size : 50,5 Mb
Release : 2017-04-07
Category : Geometry
ISBN : 9781470434793

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From Groups to Geometry and Back by Vaughn Climenhaga,Anatole Katok Pdf

Groups arise naturally as symmetries of geometric objects, and so groups can be used to understand geometry and topology. Conversely, one can study abstract groups by using geometric techniques and ultimately by treating groups themselves as geometric objects. This book explores these connections between group theory and geometry, introducing some of the main ideas of transformation groups, algebraic topology, and geometric group theory. The first half of the book introduces basic notions of group theory and studies symmetry groups in various geometries, including Euclidean, projective, and hyperbolic. The classification of Euclidean isometries leads to results on regular polyhedra and polytopes; the study of symmetry groups using matrices leads to Lie groups and Lie algebras. The second half of the book explores ideas from algebraic topology and geometric group theory. The fundamental group appears as yet another group associated to a geometric object and turns out to be a symmetry group using covering spaces and deck transformations. In the other direction, Cayley graphs, planar models, and fundamental domains appear as geometric objects associated to groups. The final chapter discusses groups themselves as geometric objects, including a gentle introduction to Gromov's theorem on polynomial growth and Grigorchuk's example of intermediate growth. The book is accessible to undergraduate students (and anyone else) with a background in calculus, linear algebra, and basic real analysis, including topological notions of convergence and connectedness. This book is a result of the MASS course in algebra at Penn State University in the fall semester of 2009.

Author : P. M. Neumann,Gabrielle A. Stoy,E. C. Thompson
Publisher : Oxford University Press, USA
Page : 268 pages
File Size : 49,5 Mb
Release : 1994
Category : Language Arts & Disciplines
ISBN : 0198534515

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Groups and Geometry by P. M. Neumann,Gabrielle A. Stoy,E. C. Thompson Pdf

Contains the Oxford Mathematical Institute notes for undergraduate and first-year postgraduates. The first half of the book covers groups, the second half covers geometry and both parts contain a number of exercises.

Author : Martin W. Liebeck
Publisher : Cambridge University Press
Page : 505 pages
File Size : 41,8 Mb
Release : 1992-09-10
Category : Mathematics
ISBN : 9780521406857

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Groups, Combinatorics and Geometry by Martin W. Liebeck Pdf

This volume contains a collection of papers on the subject of the classification of finite simple groups.

Author : R. P. Burn
Publisher : Cambridge University Press
Page : 260 pages
File Size : 54,8 Mb
Release : 1987-09-03
Category : Mathematics
ISBN : 0521347939

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Groups by R. P. Burn Pdf

Following the same successful approach as Dr. Burn's previous book on number theory, this text consists of a carefully constructed sequence of questions that will enable the reader, through participation, to study all the group theory covered by a conventional first university course. An introduction to vector spaces, leading to the study of linear groups, and an introduction to complex numbers, leading to the study of Möbius transformations and stereographic projection, are also included. Quaternions and their relationships to 3-dimensional isometries are covered, and the climax of the book is a study of the crystallographic groups, with a complete analysis of these groups in two dimensions.

Author : Anonim
Publisher : Unknown
Page : 518 pages
File Size : 54,5 Mb
Release : 2008
Category : Algebra
ISBN : UOM:39015085195405

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Algebras, Groups, and Geometries by Anonim Pdf

Author : D.J. Collins,R.I. Grigorchuk,P.F. Kurchanov,H. Zieschang
Publisher : Springer Science & Business Media
Page : 248 pages
File Size : 45,6 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9783642580130

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Algebra VII by D.J. Collins,R.I. Grigorchuk,P.F. Kurchanov,H. Zieschang Pdf

From the reviews: "... The book under review consists of two monographs on geometric aspects of group theory ... Together, these two articles form a wide-ranging survey of combinatorial group theory, with emphasis very much on the geometric roots of the subject. This will be a useful reference work for the expert, as well as providing an overview of the subject for the outsider or novice. Many different topics are described and explored, with the main results presented but not proved. This allows the interested reader to get the flavour of these topics without becoming bogged down in detail. Both articles give comprehensive bibliographies, so that it is possible to use this book as the starting point for a more detailed study of a particular topic of interest. ..." Bulletin of the London Mathematical Society, 1996

Author : Mladen Bestvina,Michah Sageev,Karen Vogtmann
Publisher : American Mathematical Soc.
Page : 417 pages
File Size : 49,6 Mb
Release : 2014-12-24
Category : Mathematics
ISBN : 9781470412272

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Geometric Group Theory by Mladen Bestvina,Michah Sageev,Karen Vogtmann Pdf

Geometric group theory refers to the study of discrete groups using tools from topology, geometry, dynamics and analysis. The field is evolving very rapidly and the present volume provides an introduction to and overview of various topics which have played critical roles in this evolution. The book contains lecture notes from courses given at the Park City Math Institute on Geometric Group Theory. The institute consists of a set of intensive short courses offered by leaders in the field, designed to introduce students to exciting, current research in mathematics. These lectures do not duplicate standard courses available elsewhere. The courses begin at an introductory level suitable for graduate students and lead up to currently active topics of research. The articles in this volume include introductions to CAT(0) cube complexes and groups, to modern small cancellation theory, to isometry groups of general CAT(0) spaces, and a discussion of nilpotent genus in the context of mapping class groups and CAT(0) groups. One course surveys quasi-isometric rigidity, others contain an exploration of the geometry of Outer space, of actions of arithmetic groups, lectures on lattices and locally symmetric spaces, on marked length spectra and on expander graphs, Property tau and approximate groups. This book is a valuable resource for graduate students and researchers interested in geometric group theory. Titles in this series are co-published with the Institute for Advanced Study/Park City Mathematics Institute. Members of the Mathematical Association of America (MAA) and the National Council of Teachers of Mathematics (NCTM) receive a 20% discount from list price.

Author : B. Rosenfeld,Bill Wiebe
Publisher : Springer Science & Business Media
Page : 414 pages
File Size : 54,7 Mb
Release : 2013-03-09
Category : Mathematics
ISBN : 9781475753257

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Geometry of Lie Groups by B. Rosenfeld,Bill Wiebe Pdf

This book is the result of many years of research in Non-Euclidean Geometries and Geometry of Lie groups, as well as teaching at Moscow State University (1947- 1949), Azerbaijan State University (Baku) (1950-1955), Kolomna Pedagogical Col lege (1955-1970), Moscow Pedagogical University (1971-1990), and Pennsylvania State University (1990-1995). My first books on Non-Euclidean Geometries and Geometry of Lie groups were written in Russian and published in Moscow: Non-Euclidean Geometries (1955) [Ro1] , Multidimensional Spaces (1966) [Ro2] , and Non-Euclidean Spaces (1969) [Ro3]. In [Ro1] I considered non-Euclidean geometries in the broad sense, as geometry of simple Lie groups, since classical non-Euclidean geometries, hyperbolic and elliptic, are geometries of simple Lie groups of classes Bn and D , and geometries of complex n and quaternionic Hermitian elliptic and hyperbolic spaces are geometries of simple Lie groups of classes An and en. [Ro1] contains an exposition of the geometry of classical real non-Euclidean spaces and their interpretations as hyperspheres with identified antipodal points in Euclidean or pseudo-Euclidean spaces, and in projective and conformal spaces. Numerous interpretations of various spaces different from our usual space allow us, like stereoscopic vision, to see many traits of these spaces absent in the usual space.

Author : A.Yu. Ol'shanskii
Publisher : Springer Science & Business Media
Page : 540 pages
File Size : 55,8 Mb
Release : 1991-10-31
Category : Mathematics
ISBN : 0792313941

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Geometry of Defining Relations in Groups by A.Yu. Ol'shanskii Pdf

The main feature of this book is a systematic application of elementary geometric and topological techniques for solving problems that arise naturally in algebra. After an account of preliminary material, there is a discussion of a geometrically intuitive interpretation of the derivation of consequences of defining relations of groups. A study is made of planar and certain other two-dimensional maps connected with well-known problems in general group theory, such as the problems of Burnside and O. Yu. Schmidt. The method of cancellation diagrams developed here is applied to these and to a series of other problems. This monograph is addressed to research workers and students in universities, and may be used as a basis for a series of specialized lectures or seminars.

Author : Christian Rosendal
Publisher : Cambridge University Press
Page : 309 pages
File Size : 49,7 Mb
Release : 2021-12-16
Category : Mathematics
ISBN : 9781108842471

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Coarse Geometry of Topological Groups by Christian Rosendal Pdf

Provides a general framework for doing geometric group theory for non-locally-compact topological groups arising in mathematical practice.

Author : Lino Di Martino,William Kantor,Guglielmo Lunardon,Antonio Pasini,Maria Clara Tamburini
Publisher : Birkhäuser
Page : 267 pages
File Size : 43,9 Mb
Release : 2013-12-01
Category : Mathematics
ISBN : 9783034888196

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Groups and Geometries by Lino Di Martino,William Kantor,Guglielmo Lunardon,Antonio Pasini,Maria Clara Tamburini Pdf

On September 1-7, 1996 a conference on Groups and Geometries took place in lovely Siena, Italy. It brought together experts and interested mathematicians from numerous countries. The scientific program centered around invited exposi tory lectures; there also were shorter research announcements, including talks by younger researchers. The conference concerned a broad range of topics in group theory and geometry, with emphasis on recent results and open problems. Special attention was drawn to the interplay between group-theoretic methods and geometric and combinatorial ones. Expanded versions of many of the talks appear in these Proceedings. This volume is intended to provide a stimulating collection of themes for a broad range of algebraists and geometers. Among those themes, represented within the conference or these Proceedings, are aspects of the following: 1. the classification of finite simple groups, 2. the structure and properties of groups of Lie type over finite and algebraically closed fields of finite characteristic, 3. buildings, and the geometry of projective and polar spaces, and 4. geometries of sporadic simple groups. We are grateful to the authors for their efforts in providing us with manuscripts in LaTeX. Barbara Priwitzer and Thomas Hintermann, Mathematics Editors of Birkhauser, have been very helpful and supportive throughout the preparation of this volume.

Author : A. Barlotti,E.W. Ellers,P. Plaumann,K. Strambach
Publisher : Springer Science & Business Media
Page : 455 pages
File Size : 53,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401133821

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Generators and Relations in Groups and Geometries by A. Barlotti,E.W. Ellers,P. Plaumann,K. Strambach Pdf

Every group is represented in many ways as an epimorphic image of a free group. It seems therefore futile to search for methods involving generators and relations which can be used to detect the structure of a group. Nevertheless, results in the indicated direction exist. The clue is to ask the right question. Classical geometry is a typical example in which the factorization of a motion into reflections or, more generally, of a collineation into central collineations, supplies valuable information on the geometric and algebraic structure. This mode of investigation has gained momentum since the end of last century. The tradition of geometric-algebraic interplay brought forward two branches of research which are documented in Parts I and II of these Proceedings. Part II deals with the theory of reflection geometry which culminated in Bachmann's work where the geometric information is encoded in properties of the group of motions expressed by relations in the generating involutions. This approach is the backbone of the classification of motion groups for the classical unitary and orthogonal planes. The axioms in this char acterization are natural and plausible. They provoke the study of consequences of subsets of axioms which also yield natural geometries whose exploration is rewarding. Bachmann's central axiom is the three reflection theorem, showing that the number of reflections needed to express a motion is of great importance.

Author : Meinolf Geck
Publisher : Oxford University Press
Page : 321 pages
File Size : 54,6 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9780199676163

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An Introduction to Algebraic Geometry and Algebraic Groups by Meinolf Geck Pdf

An accessible text introducing algebraic groups at advanced undergraduate and early graduate level, this book covers the conjugacy of Borel subgroups and maximal tori, the theory of algebraic groups with a BN-pair, Frobenius maps on affine varieties and algebraic groups, zeta functions and Lefschetz numbers for varieties over finite fields.