群作用手册

Part I: Geometric Topology

Exotic Topology in Geometry and Dynamics

F. Thomas Farrell, Andrey Gogolev, and Pedro Ontaneda

1 The space of negatively curved metrics

2 The Teichmiiller and the moduli spaces of negatively curved metrics

3 Bundles with negatively curved fibers

4 Exotic topology and hyperbolic dynamical systems

References

A Survey of Surface Braid Groups and the Lower Algebraic K-Theory of Their Group Rings

John Guaschi and Daniel Juan-Pineda

1 Introduction

2 Basic definitions of surface braid groups

3 Some properties of surface braid groups

4 Braid groups of the sphere and the projective plane

5 K-theory of surface braid groups

Acknowledgments

References

The Isomorphism Conjecture for Groups with Generalized Free Product Structure

Sayed K. Roushon

1 Introduction

2 Lower K-theory and surgery L-theory of groups acting on trees

3 The isomorphism conjecture for groups acting on trees

Acknowledgments

References

Group Actions on Banach Spaces

Piotr W. Nowak

1 Introduction

2 Preliminaries

3 Fixed points

4 Metrically proper actions

5 Final remarks

Acknowledgments

References

Topological Spherical Space Forms

Ian Hambleton

1 Introduction

2 First steps: pre-1960

3 Swan complexes: 1960-1970

4 Semicharacteristic obstructions and 3-manifolds

5 Poincare complexes

6 Surgery theory and stable existence

7 The finiteness obstruction

8 Space forms in the period dimension

9 Topological Euclidean space forms

10 Further topics

References

Part II: Representations and Deformations

Dynamics on Character Varieties: A Survey

Richard D. Canary

1 Overview

2 Basic definitions

3 Teichmuller space

4 Convex cocompact representations

5 Hitchin representations

6 Anosov representations

7 PSL(2, C)-character varieties

8 Appendix

References

Higgs Bundles and Representations of Complex Hyperbolic Lattices

Julien Maubon

1 Introduction

2 The symmetric space SV(p,q)/S(V(p) × U(q))

3 Representations, flat bundles, Higgs bundles

4 Proof of the Milnor-Wood type inequality

5 Maximal representations

References

Geometric Limits of Kleinian Groups and Their Applications

Ken'ichi Ohshika

1 Introduction

2 Preliminaries

3 Examples of geometric limits

4 Classification of geometric limits

5 Boundaries of deformation spaces

References

Deformations of Surfaces Embedded in 4-dimensional Manifolds and Their Mapping Class Groups

Susumu Hirose

1 Introduction

2 Preliminaries

3 Surfaces standardly embedded in S4

4 Flexible surfaces in 4-manifolds

5 Knotted embeddings in S4

6 Problems

Acknowledgments

References

On the Geography of Lefschetz Fibrations

Naoyuki Monden

1 Introduction

2 Lefschetz fibrations

3 Examples of Lefschetz fibrations

4 Signatures of Lefschetz fibrations

5 Fundamental groups of Lefschetz fibrations

6 Some methods to construct Lefschetz fibrations

7 Sections of Lefschetz fibrations

8 The geography problems of symplectic 4-manifolds and Lefschetz fibrations

9 Holomorphic and non-holomorphic Lefschetz fibrations

10 The slopes of Lefschetz fibrations with b+ 2= 1

Acknowledgments

References

Part III: Geometric Groups

The Modular Group and Its Actions

A. Muhammed Uludag

1 Introduction

2 Rise of the modular group

3 The bipartite Farey tree F and its boundary

4 The projective general linear group over Z

5 Action on the upper half plane H

6 Modular graphs and dessins

7 The action on binary quadratic forms

8 Et cetera

Appendix: Gosper's algorithm for continued fraction arithmetic

Acknowledgments

References

Groups with Structures and Homotopy Groups

Jie Wu

1 Introduction

2 Simplicial groups

3 Simplicial structure on geometric groups

Acknowledgments

References

Abstract Homomorphisms of Algebraic Groups and Applications

Igor A. Rapinchuk

1 Introduction

2 On algebraic rings

3 K-theoretic background

4 Rigidity results over commutative rings

5 Rigidity results over noncommutative rings

6 Applications to character varieties and deformations of representations

Acknowledgments

References

Part IV: Geometric Invariants and Growth of Discrete Groups

Some Topics in Geometric Invariant Theory over Non-algebraically Closed Fields

1 Introduction

2 An overview of geometric invariant theory. Observability and related notions

3 Stability in geometric invariant theory over non-algebraically closed fields

4 Topology of relative orbits for actions of algebraic groups over completely valued fields

Acknowledgments

References

Computation of the Spherical Growth Series of Finitely Generated Groups and Monoids by Using Automata

Michihiko Fujii

1 Introduction

2 Finite-state automata and the subset construction

3 Unique-geodesic word acceptors for groups and monoids, and growth series

4 Examples of rational growth series and irrational growth series

5 The spherical growth series of Artin groups and Artin monoids

6 The pure Artin group PI2(k) and the monoid P+ I2(k) of dihedral type

7 Automata for geodesic representatives of P+ I2 (k)

8 Unique-geodesic word acceptors constructed by subset construction

9 The spherical growth series of P+ I2 (k)

10 The spherical growth series of P+ I2 (k)

Acknowledgments

References

Part V: Music and Group Actions

Mathematics and Group Theory in Music

Athanase Papadopoulos

1 Introduction

2 A brief overview of the interaction between mathematics and music

3 The music of Olivier Messiaen

4 Rhythm

5 Counterpoint

6 Modes of limited transposition

References

群和群作用是数学研究的重要对象，拥有强大的 力量并且富于美感，这可以通过它广泛出现在诸多不 同的科学领域体现出来。
　　《群作用手册》由相关领域专家撰写的一系列综 述文章组成，首次系统地展现了群作用及其应用，内 容囊括经典主题的讨论、近来的热点专业问题的论述 ，有些文章还涉及相关的历史。本书填补了数学著作 中的一项空白，适合于从初学者到相关领域专家的各 个层次读者阅读。