Author: Matthew Eugene HorakPublisher:
ISBN:
Category :
Languages : en
Pages : 248
Book Description
Mapping Class Subgroups of Outer Automorphism Groups of Free Groups
Author: Matthew Eugene HorakPublisher:
ISBN:
Category :
Languages : en
Pages : 248
Book Description
Problems on Mapping Class Groups and Related Topics
Author: Benson FarbPublisher: American Mathematical Soc.
ISBN: 0821838385
Category : Mathematics
Languages : en
Pages : 384
Book Description
The appearance of mapping class groups in mathematics is ubiquitous. The book presents 23 papers containing problems about mapping class groups, the moduli space of Riemann surfaces, Teichmuller geometry, and related areas. Each paper focusses completely on open problems and directions. The problems range in scope from specific computations, to broad programs. The goal is to have a rich source of problems which have been formulated explicitly and accessibly. The book is divided into four parts. Part I contains problems on the combinatorial and (co)homological group-theoretic aspects of mapping class groups, and the way in which these relate to problems in geometry and topology. Part II concentrates on connections with classification problems in 3-manifold theory, the theory of symplectic 4-manifolds, and algebraic geometry. A wide variety of problems, from understanding billiard trajectories to the classification of Kleinian groups, can be reduced to differential and synthetic geometry problems about moduli space. Such problems and connections are discussed in Part III. Mapping class groups are related, both concretely and philosophically, to a number of other groups, such as braid groups, lattices in semisimple Lie groups, and automorphism groups of free groups. Part IV concentrates on problems surrounding these relationships. This book should be of interest to anyone studying geometry, topology, algebraic geometry or infinite groups. It is meant to provide inspiration for everyone from graduate students to senior researchers.
Arithmetic Groups and Their Generalizations
Author: Lizhen JiPublisher: American Mathematical Soc.
ISBN: 0821848666
Category : Mathematics
Languages : en
Pages : 282
Book Description
In one guise or another, many mathematicians are familiar with certain arithmetic groups, such as $\mathbf{Z}$ or $\textrm{SL}(n, \mathbf{Z})$. Yet, many applications of arithmetic groups and many connections to other subjects within mathematics are less well known. Indeed, arithmetic groups admit many natural and important generalizations. The purpose of this expository book is to explain, through some brief and informal comments and extensive references, what arithmetic groups and their generalizations are, why they are important to study, and how they can be understood and applied to many fields, such as analysis, geometry, topology, number theory, representation theory, and algebraic geometry. It is hoped that such an overview will shed a light on the important role played by arithmetic groups in modern mathematics. Titles in this series are co-published with International Press, Cambridge, MA.Table of Contents: Introduction; General comments on references; Examples of basic arithmetic groups; General arithmetic subgroups and locally symmetric spaces; Discrete subgroups of Lie groups and arithmeticity of lattices in Lie groups; Different completions of $\mathbb{Q}$ and $S$-arithmetic groups over number fields; Global fields and $S$-arithmetic groups over function fields; Finiteness properties of arithmetic and $S$-arithmetic groups; Symmetric spaces, Bruhat-Tits buildings and their arithmetic quotients; Compactifications of locally symmetric spaces; Rigidity of locally symmetric spaces; Automorphic forms and automorphic representations for general arithmetic groups; Cohomology of arithmetic groups; $K$-groups of rings of integers and $K$-groups of group rings; Locally homogeneous manifolds and period domains; Non-cofinite discrete groups, geometrically finite groups; Large scale geometry of discrete groups; Tree lattices; Hyperbolic groups; Mapping class groups and outer automorphism groups of free groups; Outer automorphism group of free groups and the outer spaces; References; Index. Review from Mathematical Reviews: ...the author deserves credit for having done the tremendous job of encompassing every aspect of arithmetic groups visible in today's mathematics in a systematic manner; the book should be an important guide for some time to come.(AMSIP/43.
Subgroup Decomposition in Out(Fn)
Author: Michael HandelPublisher: American Mathematical Soc.
ISBN: 1470441136
Category : Education
Languages : en
Pages : 290
Book Description
In this work the authors develop a decomposition theory for subgroups of Out(Fn) which generalizes the decomposition theory for individual elements of Out(Fn) found in the work of Bestvina, Feighn, and Handel, and which is analogous to the decomposition theory for subgroups of mapping class groups found in the work of Ivanov.
Representation Theory
Author: Zongzhu LinPublisher: American Mathematical Soc.
ISBN: 0821845551
Category : Mathematics
Languages : en
Pages : 314
Book Description
Nothing provided
Hyperbolically Embedded Subgroups and Rotating Families in Groups Acting on Hyperbolic Spaces
Author: F. DahmaniPublisher: American Mathematical Soc.
ISBN: 1470421941
Category : Mathematics
Languages : en
Pages : 164
Book Description
he authors introduce and study the notions of hyperbolically embedded and very rotating families of subgroups. The former notion can be thought of as a generalization of the peripheral structure of a relatively hyperbolic group, while the latter one provides a natural framework for developing a geometric version of small cancellation theory. Examples of such families naturally occur in groups acting on hyperbolic spaces including hyperbolic and relatively hyperbolic groups, mapping class groups, , and the Cremona group. Other examples can be found among groups acting geometrically on spaces, fundamental groups of graphs of groups, etc. The authors obtain a number of general results about rotating families and hyperbolically embedded subgroups; although their technique applies to a wide class of groups, it is capable of producing new results even for well-studied particular classes. For instance, the authors solve two open problems about mapping class groups, and obtain some results which are new even for relatively hyperbolic groups.
Mapping Class Groups and Moduli Spaces of Riemann Surfaces
Author: Carl-Friedrich BödigheimerPublisher: American Mathematical Soc.
ISBN: 0821851675
Category : Mathematics
Languages : en
Pages : 394
Book Description
The study of mapping class groups and moduli spaces of compact Riemann surfaces is currently a central topic in topology, algebraic geometry, and conformal field theory. This book contains proceedings from two workshops held in the summer of 1991, one at the University of G\"ottingen and the other at the University of Washington at Seattle. The papers gathered here represent diverse approaches and contain several important new results. With both research and survey articles, the book appeals to mathematicians and physicists.
A Primer on Mapping Class Groups
Author: Benson FarbPublisher: Princeton University Press
ISBN: 0691147949
Category : Mathematics
Languages : en
Pages : 490
Book Description
The study of the mapping class group Mod(S) is a classical topic that is experiencing a renaissance. It lies at the juncture of geometry, topology, and group theory. This book explains as many important theorems, examples, and techniques as possible, quickly and directly, while at the same time giving full details and keeping the text nearly self-contained. The book is suitable for graduate students. A Primer on Mapping Class Groups begins by explaining the main group-theoretical properties of Mod(S), from finite generation by Dehn twists and low-dimensional homology to the Dehn-Nielsen-Baer theorem. Along the way, central objects and tools are introduced, such as the Birman exact sequence, the complex of curves, the braid group, the symplectic representation, and the Torelli group. The book then introduces Teichmüller space and its geometry, and uses the action of Mod(S) on it to prove the Nielsen-Thurston classification of surface homeomorphisms. Topics include the topology of the moduli space of Riemann surfaces, the connection with surface bundles, pseudo-Anosov theory, and Thurston's approach to the classification.
Combinatorial and Geometric Group Theory
Author: Sean ClearyPublisher: American Mathematical Soc.
ISBN: 0821828223
Category : Mathematics
Languages : en
Pages : 290
Book Description
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 compactRiemann 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.
Handbook of Teichmüller Theory
Author: Athanase PapadopoulosPublisher: European Mathematical Society
ISBN: 9783037190296
Category : Mathematics
Languages : en
Pages : 812
Book Description
The Teichmuller space of a surface was introduced by O. Teichmuller in the 1930s. It is a basic tool in the study of Riemann's moduli spaces and the mapping class groups. These objects are fundamental in several fields of mathematics, including algebraic geometry, number theory, topology, geometry, and dynamics. The original setting of Teichmuller theory is complex analysis. The work of Thurston in the 1970s brought techniques of hyperbolic geometry to the study of Teichmuller space and its asymptotic geometry. Teichmuller spaces are also studied from the point of view of the representation theory of the fundamental group of the surface in a Lie group $G$, most notably $G=\mathrm{PSL}(2,\mathbb{R})$ and $G=\mathrm{PSL}(2,\mathbb{C})$. In the 1980s, there evolved an essentially combinatorial treatment of the Teichmuller and moduli spaces involving techniques and ideas from high-energy physics, namely from string theory. The current research interests include the quantization of Teichmuller space, the Weil-Petersson symplectic and Poisson geometry of this space as well as gauge-theoretic extensions of these structures. The quantization theories can lead to new invariants of hyperbolic 3-manifolds. The purpose of this handbook is to give a panorama of some of the most important aspects of Teichmuller theory. The handbook should be useful to specialists in the field, to graduate students, and more generally to mathematicians who want to learn about the subject. All the chapters are self-contained and have a pedagogical character. They are written by leading experts in the subject.