Groups And Geometry
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Author | : Vaughn Climenhaga |
Publisher | : American Mathematical Soc. |
Total Pages | : 442 |
Release | : 2017-04-07 |
Genre | : Mathematics |
ISBN | : 1470434792 |
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 | : R. P. Burn |
Publisher | : Cambridge University Press |
Total Pages | : 260 |
Release | : 1987-09-03 |
Genre | : Mathematics |
ISBN | : 9780521347938 |
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 | : P. M. Neumann |
Publisher | : Oxford University Press, USA |
Total Pages | : 268 |
Release | : 1994 |
Genre | : Language Arts & Disciplines |
ISBN | : 9780198534518 |
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 | : Tullio Ceccherini-Silberstein |
Publisher | : Springer Nature |
Total Pages | : 468 |
Release | : 2022-01-01 |
Genre | : Mathematics |
ISBN | : 3030881091 |
This book provides a detailed exposition of a wide range of topics in geometric group theory, inspired by Gromov’s pivotal work in the 1980s. It includes classical theorems on nilpotent groups and solvable groups, a fundamental study of the growth of groups, a detailed look at asymptotic cones, and a discussion of related subjects including filters and ultrafilters, dimension theory, hyperbolic geometry, amenability, the Burnside problem, and random walks on groups. The results are unified under the common theme of Gromov’s theorem, namely that finitely generated groups of polynomial growth are virtually nilpotent. This beautiful result gave birth to a fascinating new area of research which is still active today. The purpose of the book is to collect these naturally related results together in one place, most of which are scattered throughout the literature, some of them appearing here in book form for the first time. In this way, the connections between these topics are revealed, providing a pleasant introduction to geometric group theory based on ideas surrounding Gromov's theorem. The book will be of interest to mature undergraduate and graduate students in mathematics who are familiar with basic group theory and topology, and who wish to learn more about geometric, analytic, and probabilistic aspects of infinite groups.
Author | : Roger C. Lyndon |
Publisher | : Cambridge University Press |
Total Pages | : 231 |
Release | : 1985-03-14 |
Genre | : Mathematics |
ISBN | : 0521316944 |
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 | : Clara Löh |
Publisher | : Springer |
Total Pages | : 390 |
Release | : 2017-12-19 |
Genre | : Mathematics |
ISBN | : 3319722549 |
Inspired by classical geometry, geometric group theory has in turn provided a variety of applications to geometry, topology, group theory, number theory and graph theory. This carefully written textbook provides a rigorous introduction to this rapidly evolving field whose methods have proven to be powerful tools in neighbouring fields such as geometric topology. Geometric group theory is the study of finitely generated groups via the geometry of their associated Cayley graphs. It turns out that the essence of the geometry of such groups is captured in the key notion of quasi-isometry, a large-scale version of isometry whose invariants include growth types, curvature conditions, boundary constructions, and amenability. This book covers the foundations of quasi-geometry of groups at an advanced undergraduate level. The subject is illustrated by many elementary examples, outlooks on applications, as well as an extensive collection of exercises.
Author | : Viacheslav V. Nikulin |
Publisher | : Springer Science & Business Media |
Total Pages | : 262 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 3642615708 |
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 | : Mikhail Kapranov |
Publisher | : Springer Science & Business Media |
Total Pages | : 759 |
Release | : 2008-03-05 |
Genre | : Mathematics |
ISBN | : 3764386088 |
Alexander Reznikov (1960-2003) was a brilliant and highly original mathematician. This book presents 18 articles by prominent mathematicians and is dedicated to his memory. In addition it contains an influential, so far unpublished manuscript by Reznikov of book length. The book further provides an extensive survey on Kleinian groups in higher dimensions and some articles centering on Reznikov as a person.
Author | : A.Yu. Ol'shanskii |
Publisher | : Springer Science & Business Media |
Total Pages | : 530 |
Release | : 2012-12-06 |
Genre | : Mathematics |
ISBN | : 9401136181 |
'Ht moi - ..., si favait su comment en reveniT, One service mathematics hal rendered the je n'y serais point aile.' human race. It has put C.
Author | : Pierre Anglès |
Publisher | : Springer Science & Business Media |
Total Pages | : 307 |
Release | : 2007-10-16 |
Genre | : Mathematics |
ISBN | : 0817646434 |
This book provides a self-contained overview of the role of conformal groups in geometry and mathematical physics. It features a careful development of the material, from the basics of Clifford algebras to more advanced topics. Each chapter covers a specific aspect of conformal groups and conformal spin geometry. All major concepts are introduced and followed by detailed descriptions and definitions, and a comprehensive bibliography and index round out the work. Rich in exercises that are accompanied by full proofs and many hints, the book will be ideal as a course text or self-study volume for senior undergraduates and graduate students.