Geometry, Topology and Dynamics of Character Varieties

The following sections are included:

• FOREWORD

• PREFACE

• CONTENTS

We offer a short invitation to elementary hyperbolic plane geometry. We first examine the contents of Book I of Euclid's Elements and obtain a hyperbolic plane from the Euclidean one by negating Euclid's parallel postulate and keeping all of his other axioms. Then we explore the fundamentals of hyperbolic plane geometry, and study the structure of its isometries. Finally, we obtain certain identities involving the isometries and evaluate them in the upper half-plane model to derive some trigonometric laws for hyperbolic triangles.

We give a brief introduction to hyperbolic structures on surfaces. Using the concepts of developing map and holonomy, we sketch a proof that every surface equipped with a complete hyperbolic metric is isometric to a quotient of ℍ by a Fuchsian group. We then define Teichmüller spaces and explain Fenchel-Nielsen coordinates. Finally, we introduce mapping class groups and show that they act properly discontinuously on Teichmüller space.

We give an introduction to some of the tools used in studying the degeneration of hyperbolic structures on surfaces as developed by Thurston, and in particular we describe his construction of a compactification of Teichmüller space.

We study the actions of certain groups in geometric and topological contexts. Of particular interest are free groups, various free products and right-angled Artin groups. We discuss various tools which can be used to show that a particular action is faithful. We illustrate applications of these principles in the context of hyperbolic geometry and mapping class groups of surfaces.

We consider a simple class of Kleinian groups called once punctured torus groups. In this note, we will show how to create a computer program from scratch that can visualize fundamental sets and limit sets of the groups. No knowledge of computer programming is assumed.

We discuss the relationship between the geometry of complex hyperbolic manifolds and orbifolds and the traces of elements of the corresponding subgroup of SU(2, 1). We begin by showing how geometrical information about individual isometries is encoded by their trace. We then consider traces for groups Γ of isometries in two specific cases. First, we consider the case where Γ is a free group on two generators, which we view as the fundamental group of a three holed sphere. We indicate how to use this analysis to give complex hyperbolic Fenchel-Nielsen coordinates. Secondly, we consider the case where Γ is a triangle group generated by complex reflections in three complex lines. We keep in mind similar results from the more familiar setting of Fuchsian and Kleinian groups and we explain those examples from our point of view.

Flat Lorentz (3, 1) space is the natural home for Einstein's Special Theory of Relativity, with three "space" dimensions and one "time" dimension. The geometry naturally encodes the ideas of inertial frames, time and space dilation. Much of the mathematical terminology and research interests are infused with physical considerations, but purely mathematical questions are at the core of this treatment.

This is an introduction to Lorentzian (n, 1) geometry for all n ≥ 1, examining the constant curvature spaces: flat, de Sitter (positive curvature) and anti-de Sittter spaces. The conformal boundaries of these Lorentz spacetimes will also be constructed. The connection between Lorentz (2, 1) geometry and geometry of the hyperbolic plane will also be investigated.

The Teichmüller space of a surface naturally embeds as a connected component in the moduli space of representations from the fundamental group of the surface into the group of isometries of the hyperbolic plane. We present invariants that distinguish all the connected components of the space of representations. This allows us to compute the number of connected components of these spaces both in the orientable and in the non-orientable case.

We survey recent results on surface group representations in semisimple Lie groups and local rigidity phenomena of lattices in semisimple Lie groups. We review basics about the character varieties of surface groups, its application and structures. We show that a surface group contained in a reductive real algebraic group can be deformed to become Zariski dense, unless one of the noncompact factors of the symmetric space of its Zariski closure is Hermitian of tube type. This is a kind of converse to rigidity results of Burger, Iozzi and Wienhard.

We place the representation variety in the broader context of abelian and nonabelian cohomology. We outline the equivalent constructions of the moduli spaces of flat bundles, of smooth integrable connections, and of holomorphic integrable connections over a compact Kähler manifold. In addition, we describe the moduli space of Higgs bundles and how it relates to the representation variety. We attempt to avoid abstraction, but strive to present and clarify the unifying ideas underlying the theory.