Group Actions In Ergodic Theory Geometry And Topology

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Group Actions in Ergodic Theory, Geometry, and Topology

Robert J. Zimmer
Group Actions in Ergodic Theory Geometry and Topology Book PDF

Author : Robert J. Zimmer
Publisher : University of Chicago Press
Page : 724 pages
File Size : 44,8 Mb
Release : 2019-12-23
Category : Mathematics
ISBN : 9780226568270

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Group Actions in Ergodic Theory, Geometry, and Topology by Robert J. Zimmer Pdf

Robert J. Zimmer is best known in mathematics for the highly influential conjectures and program that bear his name. Group Actions in Ergodic Theory, Geometry, and Topology: Selected Papers brings together some of the most significant writings by Zimmer, which lay out his program and contextualize his work over the course of his career. Zimmer’s body of work is remarkable in that it involves methods from a variety of mathematical disciplines, such as Lie theory, differential geometry, ergodic theory and dynamical systems, arithmetic groups, and topology, and at the same time offers a unifying perspective. After arriving at the University of Chicago in 1977, Zimmer extended his earlier research on ergodic group actions to prove his cocycle superrigidity theorem which proved to be a pivotal point in articulating and developing his program. Zimmer’s ideas opened the door to many others, and they continue to be actively employed in many domains related to group actions in ergodic theory, geometry, and topology. In addition to the selected papers themselves, this volume opens with a foreword by David Fisher, Alexander Lubotzky, and Gregory Margulis, as well as a substantial introductory essay by Zimmer recounting the course of his career in mathematics. The volume closes with an afterword by Fisher on the most recent developments around the Zimmer program.

Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces

M. Bachir Bekka,Matthias Mayer
Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces Book PDF

Author : M. Bachir Bekka,Matthias Mayer
Publisher : Cambridge University Press
Page : 214 pages
File Size : 55,8 Mb
Release : 2000-05-11
Category : Mathematics
ISBN : 0521660300

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Ergodic Theory and Topological Dynamics of Group Actions on Homogeneous Spaces by M. Bachir Bekka,Matthias Mayer Pdf

This book, first published in 2000, focuses on developments in the study of geodesic flows on homogenous spaces.

Geometry, Rigidity, and Group Actions

Robert J Zimmer
Geometry Rigidity and Group Actions Book PDF

Author : Robert J Zimmer
Publisher : University of Chicago Press
Page : 600 pages
File Size : 54,6 Mb
Release : 2011-04-15
Category : Mathematics
ISBN : 9780226237909

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Geometry, Rigidity, and Group Actions by Robert J Zimmer Pdf

The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic fields. A central development in the last fifty years is the phenomenon of rigidity, whereby one can classify actions of certain groups, such as lattices in semi-simple Lie groups. This provides a way to classify all possible symmetries of important spaces and all spaces admitting given symmetries. Paradigmatic results can be found in the seminal work of George Mostow, Gergory Margulis, and Robert J. Zimmer, among others. The papers in Geometry, Rigidity, and Group Actions explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. In some cases, the dynamics of the possible group actions are the principal focus of inquiry. In other cases, the dynamics of group actions are a tool for proving theorems about algebra, geometry, or topology. This volume contains surveys of some of the main directions in the field, as well as research articles on topics of current interest.

Geometric and Ergodic Aspects of Group Actions

S. G. Dani,Anish Ghosh
Geometric and Ergodic Aspects of Group Actions Book PDF

Author : S. G. Dani,Anish Ghosh
Publisher : Springer Nature
Page : 176 pages
File Size : 48,7 Mb
Release : 2020-01-13
Category : Mathematics
ISBN : 9789811506833

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Geometric and Ergodic Aspects of Group Actions by S. G. Dani,Anish Ghosh Pdf

This book gathers papers on recent advances in the ergodic theory of group actions on homogeneous spaces and on geometrically finite hyperbolic manifolds presented at the workshop “Geometric and Ergodic Aspects of Group Actions,” organized by the Tata Institute of Fundamental Research, Mumbai, India, in 2018. Written by eminent scientists, and providing clear, detailed accounts of various topics at the interface of ergodic theory, the theory of homogeneous dynamics, and the geometry of hyperbolic surfaces, the book is a valuable resource for researchers and advanced graduate students in mathematics.

Ergodic Theory, Groups, and Geometry

Robert J. Zimmer,Dave Witte Morris
Ergodic Theory Groups and Geometry Book PDF

Author : Robert J. Zimmer,Dave Witte Morris
Publisher : American Mathematical Soc.
Page : 87 pages
File Size : 55,9 Mb
Release : 2008
Category : Mathematics
ISBN : 9780821809808

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Ergodic Theory, Groups, and Geometry by Robert J. Zimmer,Dave Witte Morris Pdf

This introduction to ergodic theory provides an overview of important methods, major developments and open problems in the subject. The lectures in the book include additional comments at the end of each chapter with references to recent developments. These updates can help lead the graduate student to cutting-edge results in the field.

Dynamics, Ergodic Theory and Geometry

Boris Hasselblatt
Dynamics Ergodic Theory and Geometry Book PDF

Author : Boris Hasselblatt
Publisher : Cambridge University Press
Page : 324 pages
File Size : 40,8 Mb
Release : 2007-09-24
Category : Mathematics
ISBN : 9780521875417

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Dynamics, Ergodic Theory and Geometry by Boris Hasselblatt Pdf

Based on the subjects from the Clay Mathematics Institute/Mathematical Sciences Research Institute Workshop titled 'Recent Progress in Dynamics' in September and October 2004, this volume contains surveys and research articles by leading experts in several areas of dynamical systems that have experienced substantial progress. One of the major surveys is on symplectic geometry, which is closely related to classical mechanics and an exciting addition to modern geometry. The survey on local rigidity of group actions gives a broad and up-to-date account of another flourishing subject. Other papers cover hyperbolic, parabolic, and symbolic dynamics as well as ergodic theory. Students and researchers in dynamical systems, geometry, and related areas will find this book fascinating. The book also includes a fifty-page commented problem list that takes the reader beyond the areas covered by the surveys, to inspire and guide further research.

Global Aspects of Ergodic Group Actions

A. S. Kechris
Global Aspects of Ergodic Group Actions Book PDF

Author : A. S. Kechris
Publisher : American Mathematical Soc.
Page : 258 pages
File Size : 54,8 Mb
Release : 2010
Category : Mathematics
ISBN : 9780821848944

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Global Aspects of Ergodic Group Actions by A. S. Kechris Pdf

A study of ergodic, measure preserving actions of countable discrete groups on standard probability spaces. It explores a direction that emphasizes a global point of view, concentrating on the structure of the space of measure preserving actions of a given group and its associated cocycle spaces.

Geometry, Rigidity, and Group Actions

Robert J. Zimmer,Benson Farb,David Fisher
Geometry Rigidity and Group Actions Book PDF

Author : Robert J. Zimmer,Benson Farb,David Fisher
Publisher : University of Chicago Press
Page : 659 pages
File Size : 50,7 Mb
Release : 2011-04-15
Category : Mathematics
ISBN : 9780226237893

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Geometry, Rigidity, and Group Actions by Robert J. Zimmer,Benson Farb,David Fisher Pdf

The study of group actions is more than 100 years old but remains a widely studied topic in a variety of mathematic fields. A central development in the last 50 years is the phenomenon of rigidity, whereby one can classify actions of certain groups. This book looks at rigidity.

Ergodic Theory and Semisimple Groups

R.J. Zimmer
Ergodic Theory and Semisimple Groups Book PDF

Author : R.J. Zimmer
Publisher : Springer Science & Business Media
Page : 219 pages
File Size : 41,6 Mb
Release : 2013-03-14
Category : Mathematics
ISBN : 9781468494884

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Ergodic Theory and Semisimple Groups by R.J. Zimmer Pdf

This book is based on a course given at the University of Chicago in 1980-81. As with the course, the main motivation of this work is to present an accessible treatment, assuming minimal background, of the profound work of G. A. Margulis concerning rigidity, arithmeticity, and structure of lattices in semi simple groups, and related work of the author on the actions of semisimple groups and their lattice subgroups. In doing so, we develop the necessary prerequisites from earlier work of Borel, Furstenberg, Kazhdan, Moore, and others. One of the difficulties involved in an exposition of this material is the continuous interplay between ideas from the theory of algebraic groups on the one hand and ergodic theory on the other. This, of course, is not so much a mathematical difficulty as a cultural one, as the number of persons comfortable in both areas has not traditionally been large. We hope this work will also serve as a contribution towards improving that situation. While there are a number of satisfactory introductory expositions of the ergodic theory of integer or real line actions, there is no such exposition of the type of ergodic theoretic results with which we shall be dealing (concerning actions of more general groups), and hence we have assumed absolutely no knowledge of ergodic theory (not even the definition of "ergodic") on the part of the reader. All results are developed in full detail.

Dynamical Systems and Group Actions

Lewis Bowen,R. I. Grigorchuk,Yaroslav Vorobets
Dynamical Systems and Group Actions Book PDF

Author : Lewis Bowen,R. I. Grigorchuk,Yaroslav Vorobets
Publisher : American Mathematical Soc.
Page : 280 pages
File Size : 54,5 Mb
Release : 2012
Category : Mathematics
ISBN : 9780821869222

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Dynamical Systems and Group Actions by Lewis Bowen,R. I. Grigorchuk,Yaroslav Vorobets Pdf

This volume contains cutting-edge research from leading experts in ergodic theory, dynamical systems and group actions. A large part of the volume addresses various aspects of ergodic theory of general group actions including local entropy theory, universal minimal spaces, minimal models and rank one transformations. Other papers deal with interval exchange transformations, hyperbolic dynamics, transfer operators, amenable actions and group actions on graphs.

Group Actions on Manifolds

Reinhard Schultz
Group Actions on Manifolds Book PDF

Author : Reinhard Schultz
Publisher : American Mathematical Soc.
Page : 568 pages
File Size : 41,7 Mb
Release : 1985
Category : Mathematics
ISBN : 9780821850381

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Group Actions on Manifolds by Reinhard Schultz Pdf

Not merely an account of new results, this book is also a guide to motivation behind present work and potential future developments. Readers can obtain an overall understanding of the sorts of problems one studies in group actions and the methods used to study such problems. The book will be accessible to advanced graduate students who have had the equivalent of three semesters of graduate courses in topology; some previous acquaintance with the fundamentals of transformation groups is also highly desirable. The articles in this book are mainly based upon lectures at the 1983 AMS-IMS-SIAM Joint Summer Research Conference, Group Actions on Manifolds, held at the University of Colorado. A major objective was to provide an overall account of current knowledge in transformation groups; a number of survey articles describe the present state of the subject from several complementary perspectives. The book also contains some research articles, generally dealing with results presented at the conference. Finally, there is a discussion of current problems on group actions and an acknowledgment of the work and influence of D. Montgomery on the subject.

Ergodic Theoretic Methods in Group Homology

Clara Löh
Ergodic Theoretic Methods in Group Homology Book PDF

Author : Clara Löh
Publisher : Springer Nature
Page : 114 pages
File Size : 52,6 Mb
Release : 2020-03-14
Category : Mathematics
ISBN : 9783030442200

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Ergodic Theoretic Methods in Group Homology by Clara Löh Pdf

This book offers a concise introduction to ergodic methods in group homology, with a particular focus on the computation of L2-Betti numbers. Group homology integrates group actions into homological structure. Coefficients based on probability measure preserving actions combine ergodic theory and homology. An example of such an interaction is provided by L2-Betti numbers: these invariants can be understood in terms of group homology with coefficients related to the group von Neumann algebra, via approximation by finite index subgroups, or via dynamical systems. In this way, L2-Betti numbers lead to orbit/measure equivalence invariants and measured group theory helps to compute L2-Betti numbers. Similar methods apply also to compute the rank gradient/cost of groups as well as the simplicial volume of manifolds. This book introduces L2-Betti numbers of groups at an elementary level and then develops the ergodic point of view, emphasising the connection with approximation phenomena for homological gradient invariants of groups and spaces. The text is an extended version of the lecture notes for a minicourse at the MSRI summer graduate school “Random and arithmetic structures in topology” and thus accessible to the graduate or advanced undergraduate students. Many examples and exercises illustrate the material.

Foliations: Dynamics, Geometry and Topology

Masayuki Asaoka,Aziz El Kacimi Alaoui,Steven Hurder,Ken Richardson
Foliations Dynamics Geometry and Topology Book PDF

Author : Masayuki Asaoka,Aziz El Kacimi Alaoui,Steven Hurder,Ken Richardson
Publisher : Springer
Page : 198 pages
File Size : 41,6 Mb
Release : 2014-10-07
Category : Mathematics
ISBN : 9783034808712

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Foliations: Dynamics, Geometry and Topology by Masayuki Asaoka,Aziz El Kacimi Alaoui,Steven Hurder,Ken Richardson Pdf

This book is an introduction to several active research topics in Foliation Theory and its connections with other areas. It contains expository lectures showing the diversity of ideas and methods converging in the study of foliations. The lectures by Aziz El Kacimi Alaoui provide an introduction to Foliation Theory with emphasis on examples and transverse structures. Steven Hurder's lectures apply ideas from smooth dynamical systems to develop useful concepts in the study of foliations: limit sets and cycles for leaves, leafwise geodesic flow, transverse exponents, Pesin Theory and hyperbolic, parabolic and elliptic types of foliations. The lectures by Masayuki Asaoka compute the leafwise cohomology of foliations given by actions of Lie groups, and apply it to describe deformation of those actions. In his lectures, Ken Richardson studies the properties of transverse Dirac operators for Riemannian foliations and compact Lie group actions, and explains a recently proved index formula. Besides students and researchers of Foliation Theory, this book will be interesting for mathematicians interested in the applications to foliations of subjects like Topology of Manifolds, Differential Geometry, Dynamics, Cohomology or Global Analysis.

Structure and Regularity of Group Actions on One-Manifolds

Sang-hyun Kim,Thomas Koberda
Structure and Regularity of Group Actions on One Manifolds Book PDF

Author : Sang-hyun Kim,Thomas Koberda
Publisher : Springer Nature
Page : 323 pages
File Size : 48,7 Mb
Release : 2021-11-19
Category : Mathematics
ISBN : 9783030890063

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Structure and Regularity of Group Actions on One-Manifolds by Sang-hyun Kim,Thomas Koberda Pdf

This book presents the theory of optimal and critical regularities of groups of diffeomorphisms, from the classical work of Denjoy and Herman, up through recent advances. Beginning with an investigation of regularity phenomena for single diffeomorphisms, the book goes on to describes a circle of ideas surrounding Filipkiewicz's Theorem, which recovers the smooth structure of a manifold from its full diffeomorphism group. Topics covered include the simplicity of homeomorphism groups, differentiability of continuous Lie group actions, smooth conjugation of diffeomorphism groups, and the reconstruction of spaces from group actions. Various classical and modern tools are developed for controlling the dynamics of general finitely generated group actions on one-dimensional manifolds, subject to regularity bounds, including material on Thompson's group F, nilpotent groups, right-angled Artin groups, chain groups, finitely generated groups with prescribed critical regularities, and applications to foliation theory and the study of mapping class groups. The book will be of interest to researchers in geometric 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 : 44,9 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.