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
