Cover

Title page

Contents

Foreword

Preface

Complex Kleinian groups

Introduction

1. Kleinian groups in real hyperbolic space

2. Complex Kleinian groups

3. On the classification of projective automorphisms

4. Dynamics in complex dimension 2

5. Geometry in complex dimension 2

References

Complete Lorentzian 3-manifolds

1. Introduction

2. Basic Lorentzian geometry

3. Proper actions and locally homogeneous Lorentzian 3-manifolds

4. Affine deformations and the Margulis invariant

5. Einstein Universe

References

The Goldman bracket and the intersection of curves on surfaces

1. Introduction

2. The \Z-module of curves

3. The Goldman bracket

4. Examples

5. Relation between the bracket of two curves and the number of intersection points.

6. Open problems and generalizations

References

An introduction to flows on homogeneous spaces

1. Homogeneous spaces

2. Measures on homogeneous spaces

3. Ergodic properties

4. Unitary representations

5. Flows on 𝑆𝐿(2,\R)/Γ

6. Flows on 𝑆𝐿(𝑛,\R)/Γ

7. Duality

8. Diophantine approximation

9. Geodesic flows and applications

References

Quantitative Geometry of Hyperbolic Manifolds

1. The Banach-Tarski paradox

2. Uniform restrictions on discrete groups

3. Quantitative Mostow rigidity

4. Diameter and homology

References

Discrete Groups and Riemann Surfaces

1. Prerequisites

2. Riemann surfaces

3. Holomorphic maps

4. Galois groups

5. Uniformization

6. Greenberg-Singerman extensions

7. Further reading

References

A Note on Chern’s Theorem on Invariant Measures

1. Introduction

2. Chern’s Criterion for existence of Invariant measures

3. A New approach to Chern’s Theorem

4. Cauchy-Crofton Type Formulae

References

Random Affine Lattices

References

Upper central series for the group of unitriangular automorphisms of a free associative algebra

1. Introduction

2. Some previous results and remark

3. Unitriangular group 𝑈₂

4. Unitriangular group 𝑈₃

5. Center of the unitriangular group 𝑈_{𝑛}, 𝑛≥4

References

Hermitian structure and bundles on 𝐺/Γ

1. Introduction

2. Hermitian structure on 𝐺/Γ

3. Automorphisms of the pair (𝐺,Γ)

4. Homogeneous bundles on 𝐺/Γ

5. Extendable homomorphisms from the lattice

References

Log-Riemann surfaces, Caratheodory convergence and Euler’s formula

1. Introduction

1.1. Acknowledgements

2. Log-Riemann surfaces

2.1. Definitions

2.2. Examples

3. Caratheodory Theorem for log-Riemann surfaces

3.1. Caratheodory convergence of log-Riemann surfaces

3.2. Convergence of uniformizations

References

Uniformization of simply connected finite type Log-Riemann surfaces

1. Introduction

2. Cell decompositions of log-Riemann surfaces

2.1. Decomposition into stars

2.2. The skeleton and fundamental group

2.3. Truncation and approximation by finite sheeted surfaces

2.4. Compactness for uniformly finite type log-Riemann surfaces

2.5. Decomposition into Kobayashi-Nevanlinna cells

2.6. Kobayashi-Nevanlinna parabolicity criterion

3. Uniformization theorems

References

The Euler Characteristic of a Haken 4-Manifold

1. Introduction

2. Basic Facts about Haken 𝑛-Manifolds

3. Approach to the Sign Conjecture for Haken 4-Manifolds

4. Results

5. Determination of 𝜙-function and the transformation law

6. Evaluation of the 𝜙-function on Haken 4-cells

7. Concluding Discussion

Acknowledgements

References

A discreteness criterion for groups containing parabolic isometries

1. Preliminaries

2. The Criterion

3. The 4-dimensional case

References

On the 𝑧-classes in a centrally finite division ring

1. Introduction

2. Some Generalities on 𝑧-Classes

3. 𝑧-Classes in Centrally Finite Division Rings

4. Wedderburn’s Theorem on Finite Division Rings

References

On Lorentz spacetimes of constant curvature

1. Introduction

2. The pseudo-Riemannian geometry of 𝐺

3. Contracting maps and properly discontinuous actions

4. Main results

References

On the asymptotic behavior of complex earthquakes and Teichmüller disks

1. Introduction

2. Background

3. Preliminaries

4. Proof of Theorem 1.1

5. Plumbing disks and Theorem 1.2

References

Characteristically Simple Beauville Groups, I: Cartesian Powers of Alternating Groups

1. Introduction

2. Beauville surfaces and structures

3. Generating cartesian powers

4. Evaluating 𝜑₂(𝐻), 𝑑₂(𝐻) and 𝑐₂(𝐻)

5. Beauville structures in cartesian powers

6. Primitive permutation groups

7. Small alternating groups

8. Larger alternating groups

References

Relatively Hyperbolic Spaces

1. Introduction

2. Hyperbolicity and Nearest Point Projections

3. Electric Geometry

4. Application

Acknowledgements

References

Complex hyperbolic free groups with many parabolic elements

1. Introduction

2. Fixed point tetrahedra of thrice punctured sphere groups

3. Constructing thrice punctured sphere groups from tetrahedra

4. Thrice punctured sphere groups with a three-fold symmetry.

5. Discreteness

References

On Fatou components and omitted values

1. Introduction

2. Implications of an omitted value

3. The Result and its Proof

4. Examples

References

Some dynamical properties of certain meromorphic functions

1. Introduction

2. Notations

3. Basic Dynamical Properties of 𝑆_{𝜆}

4. The Dynamic Planes of the Function 𝑓∈𝐹

References

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