Geometric Group Theory

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Geometric Group Theory

Clara Löh
Geometric Group Theory Book PDF

Author : Clara Löh
Publisher : Springer
Page : 389 pages
File Size : 51,8 Mb
Release : 2017-12-19
Category : Mathematics
ISBN : 9783319722542

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Geometric Group Theory by Clara Löh Pdf

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.

Office Hours with a Geometric Group Theorist

Matt Clay,Dan Margalit
Office Hours with a Geometric Group Theorist Book PDF

Author : Matt Clay,Dan Margalit
Publisher : Princeton University Press
Page : 456 pages
File Size : 53,6 Mb
Release : 2017-07-11
Category : Mathematics
ISBN : 9781400885398

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Office Hours with a Geometric Group Theorist by Matt Clay,Dan Margalit Pdf

Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. Office Hours with a Geometric Group Theorist brings together leading experts who provide one-on-one instruction on key topics in this exciting and relatively new field of mathematics. It's like having office hours with your most trusted math professors. An essential primer for undergraduates making the leap to graduate work, the book begins with free groups—actions of free groups on trees, algorithmic questions about free groups, the ping-pong lemma, and automorphisms of free groups. It goes on to cover several large-scale geometric invariants of groups, including quasi-isometry groups, Dehn functions, Gromov hyperbolicity, and asymptotic dimension. It also delves into important examples of groups, such as Coxeter groups, Thompson's groups, right-angled Artin groups, lamplighter groups, mapping class groups, and braid groups. The tone is conversational throughout, and the instruction is driven by examples. Accessible to students who have taken a first course in abstract algebra, Office Hours with a Geometric Group Theorist also features numerous exercises and in-depth projects designed to engage readers and provide jumping-off points for research projects.

Topics in Geometric Group Theory

Pierre de la Harpe
Topics in Geometric Group Theory Book PDF

Author : Pierre de la Harpe
Publisher : University of Chicago Press
Page : 320 pages
File Size : 40,5 Mb
Release : 2000-10-15
Category : Education
ISBN : 0226317196

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Topics in Geometric Group Theory by Pierre de la Harpe Pdf

In this book, Pierre de la Harpe provides a concise and engaging introduction to geometric group theory, a new method for studying infinite groups via their intrinsic geometry that has played a major role in mathematics over the past two decades. A recognized expert in the field, de la Harpe adopts a hands-on approach, illustrating key concepts with numerous concrete examples. The first five chapters present basic combinatorial and geometric group theory in a unique and refreshing way, with an emphasis on finitely generated versus finitely presented groups. In the final three chapters, de la Harpe discusses new material on the growth of groups, including a detailed treatment of the "Grigorchuk group." Most sections are followed by exercises and a list of problems and complements, enhancing the book's value for students; problems range from slightly more difficult exercises to open research problems in the field. An extensive list of references directs readers to more advanced results as well as connections with other fields.

Geometric Group Theory

Cornelia Druţu,Michael Kapovich
Geometric Group Theory Book PDF

Author : Cornelia Druţu,Michael Kapovich
Publisher : American Mathematical Soc.
Page : 819 pages
File Size : 42,7 Mb
Release : 2018-03-28
Category : Geometric group theory
ISBN : 9781470411046

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Geometric Group Theory by Cornelia Druţu,Michael Kapovich Pdf

The key idea in geometric group theory is to study infinite groups by endowing them with a metric and treating them as geometric spaces. This applies to many groups naturally appearing in topology, geometry, and algebra, such as fundamental groups of manifolds, groups of matrices with integer coefficients, etc. The primary focus of this book is to cover the foundations of geometric group theory, including coarse topology, ultralimits and asymptotic cones, hyperbolic groups, isoperimetric inequalities, growth of groups, amenability, Kazhdan's Property (T) and the Haagerup property, as well as their characterizations in terms of group actions on median spaces and spaces with walls. The book contains proofs of several fundamental results of geometric group theory, such as Gromov's theorem on groups of polynomial growth, Tits's alternative, Stallings's theorem on ends of groups, Dunwoody's accessibility theorem, the Mostow Rigidity Theorem, and quasiisometric rigidity theorems of Tukia and Schwartz. This is the first book in which geometric group theory is presented in a form accessible to advanced graduate students and young research mathematicians. It fills a big gap in the literature and will be used by researchers in geometric group theory and its applications.

Combinatorial and Geometric Group Theory

Sean Cleary,Stephen Berman,Robert Gilman,Alexei G. Myasnikov,Vladimir Shpilrain
Combinatorial and Geometric Group Theory Book PDF

Author : Sean Cleary,Stephen Berman,Robert Gilman,Alexei G. Myasnikov,Vladimir Shpilrain
Publisher : American Mathematical Soc.
Page : 275 pages
File Size : 41,9 Mb
Release : 2002
Category : Mathematics
ISBN : 9780821828229

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Combinatorial and Geometric Group Theory by Sean Cleary,Stephen Berman,Robert Gilman,Alexei G. Myasnikov,Vladimir Shpilrain Pdf

This volume grew out of two AMS conferences held at Columbia University (New York, NY) and the Stevens Institute of Technology (Hoboken, NJ) and presents articles on a wide variety of topics in group theory. Readers will find a variety of contributions, including a collection of over 170 open problems in combinatorial group theory, three excellent survey papers (on boundaries of hyperbolic groups, on fixed points of free group automorphisms, and on groups of automorphisms of compact Riemann surfaces), and several original research papers that represent the diversity of current trends in combinatorial and geometric group theory. The book is an excellent reference source for graduate students and research mathematicians interested in various aspects of group theory.

Geometric Group Theory

Mladen Bestvina,Michah Sageev,Karen Vogtmann
Geometric Group Theory Book PDF

Author : Mladen Bestvina,Michah Sageev,Karen Vogtmann
Publisher : American Mathematical Soc.
Page : 339 pages
File Size : 51,6 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.

Combinatorial and Geometric Group Theory

Oleg Bogopolski,Inna Bumagin,Olga Kharlampovich,Enric Ventura
Combinatorial and Geometric Group Theory Book PDF

Author : Oleg Bogopolski,Inna Bumagin,Olga Kharlampovich,Enric Ventura
Publisher : Springer Science & Business Media
Page : 315 pages
File Size : 53,8 Mb
Release : 2011-01-28
Category : Mathematics
ISBN : 9783764399115

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Combinatorial and Geometric Group Theory by Oleg Bogopolski,Inna Bumagin,Olga Kharlampovich,Enric Ventura Pdf

This volume assembles several research papers in all areas of geometric and combinatorial group theory originated in the recent conferences in Dortmund and Ottawa in 2007. It contains high quality refereed articles developing new aspects of these modern and active fields in mathematics. It is also appropriate to advanced students interested in recent results at a research level.

Topics in Groups and Geometry

Tullio Ceccherini-Silberstein,Michele D'Adderio
Topics in Groups and Geometry Book PDF

Author : Tullio Ceccherini-Silberstein,Michele D'Adderio
Publisher : Springer Nature
Page : 468 pages
File Size : 45,7 Mb
Release : 2022-01-01
Category : Mathematics
ISBN : 9783030881092

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Topics in Groups and Geometry by Tullio Ceccherini-Silberstein,Michele D'Adderio Pdf

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.

Topological Methods in Group Theory

Ross Geoghegan
Topological Methods in Group Theory Book PDF

Author : Ross Geoghegan
Publisher : Springer Science & Business Media
Page : 473 pages
File Size : 52,5 Mb
Release : 2007-12-17
Category : Mathematics
ISBN : 9780387746111

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Topological Methods in Group Theory by Ross Geoghegan Pdf

This book is about the interplay between algebraic topology and the theory of infinite discrete groups. It is a hugely important contribution to the field of topological and geometric group theory, and is bound to become a standard reference in the field. To keep the length reasonable and the focus clear, the author assumes the reader knows or can easily learn the necessary algebra, but wants to see the topology done in detail. The central subject of the book is the theory of ends. Here the author adopts a new algebraic approach which is geometric in spirit.

Geometry of Defining Relations in Groups

A.Yu. Ol'shanskii
Geometry of Defining Relations in Groups Book PDF

Author : A.Yu. Ol'shanskii
Publisher : Springer Science & Business Media
Page : 530 pages
File Size : 46,8 Mb
Release : 2012-12-06
Category : Mathematics
ISBN : 9789401136181

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

'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.

From Groups to Geometry and Back

Vaughn Climenhaga,Anatole Katok
From Groups to Geometry and Back Book PDF

Author : Vaughn Climenhaga,Anatole Katok
Publisher : American Mathematical Soc.
Page : 420 pages
File Size : 41,9 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.

Geometric Set Theory

Paul B. Larson,Jindrich Zapletal
Geometric Set Theory Book PDF

Author : Paul B. Larson,Jindrich Zapletal
Publisher : American Mathematical Soc.
Page : 330 pages
File Size : 51,6 Mb
Release : 2020-07-16
Category : Education
ISBN : 9781470454623

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Geometric Set Theory by Paul B. Larson,Jindrich Zapletal Pdf

This book introduces a new research direction in set theory: the study of models of set theory with respect to their extensional overlap or disagreement. In Part I, the method is applied to isolate new distinctions between Borel equivalence relations. Part II contains applications to independence results in Zermelo–Fraenkel set theory without Axiom of Choice. The method makes it possible to classify in great detail various paradoxical objects obtained using the Axiom of Choice; the classifying criterion is a ZF-provable implication between the existence of such objects. The book considers a broad spectrum of objects from analysis, algebra, and combinatorics: ultrafilters, Hamel bases, transcendence bases, colorings of Borel graphs, discontinuous homomorphisms between Polish groups, and many more. The topic is nearly inexhaustible in its variety, and many directions invite further investigation.

Topology and Geometric Group Theory

Michael W. Davis,James Fowler,Jean-François Lafont,Ian J. Leary
Topology and Geometric Group Theory Book PDF

Author : Michael W. Davis,James Fowler,Jean-François Lafont,Ian J. Leary
Publisher : Springer
Page : 174 pages
File Size : 55,8 Mb
Release : 2018-06-14
Category : Mathematics
ISBN : 3319828835

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Topology and Geometric Group Theory by Michael W. Davis,James Fowler,Jean-François Lafont,Ian J. Leary Pdf

This book presents articles at the interface of two active areas of research: classical topology and the relatively new field of geometric group theory. It includes two long survey articles, one on proofs of the Farrell–Jones conjectures, and the other on ends of spaces and groups. In 2010–2011, Ohio State University (OSU) hosted a special year in topology and geometric group theory. Over the course of the year, there were seminars, workshops, short weekend conferences, and a major conference out of which this book resulted. Four other research articles complement these surveys, making this book ideal for graduate students and established mathematicians interested in entering this area of research.

Geometric Group Theory: Volume 1

Graham A. Niblo,Martin A. Roller
Geometric Group Theory Volume 1 Book PDF

Author : Graham A. Niblo,Martin A. Roller
Publisher : Cambridge University Press
Page : 226 pages
File Size : 50,7 Mb
Release : 1993-07-30
Category : Mathematics
ISBN : 9780521435291

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Geometric Group Theory: Volume 1 by Graham A. Niblo,Martin A. Roller Pdf

For anyone whose interest lies in the interplay between groups and geometry, these books will be an essential addition to their library.

The Geometry and Topology of Coxeter Groups

Michael Davis
The Geometry and Topology of Coxeter Groups Book PDF

Author : Michael Davis
Publisher : Princeton University Press
Page : 601 pages
File Size : 51,8 Mb
Release : 2008
Category : Mathematics
ISBN : 9780691131382

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The Geometry and Topology of Coxeter Groups by Michael Davis Pdf

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.