Geometric and Ergodic Aspects of Group Actions (Infosys Science Foundation Series)

Preface
Contents
About the Editors
Lectures on Kleinian Groups
	1 Geometric Structures on Surfaces: Fuchsian Groups
	2 Kleinian Groups
		2.1 Algebraic and Geometric Topology on Space  of Representations
		2.2 The Hyperbolic Geometry Point of View
		2.3 Asymptotic/Complex Dynamic Viewpoint from
	3 Degenerate Representations
		3.1 Convex Cores
	4 Geodesic Laminations
		4.1 Pleating Locus
		4.2 Ending Laminations
	5 The Ending Lamination Theorem
	6 Cannon–Thurston Maps
		6.1 Motivating Questions
		6.2 Three-Manifolds Fibering Over the Circle
		6.3 Geometric Group Theory Proof
		6.4 Cannon–Thurston Maps for Kleinian Surface Groups
	7 Generalizations and Applications: Kleinian Groups
		7.1 Finitely Generated Kleinian Groups
		7.2 Cannon–Thurston Maps and the Ending Lamination Theorem
		7.3 Motions of Limit Sets
	References
Horocycle Flows on Surfaces  with Infinite Genus
	1 Introduction
	2 Classification of Invariant Radon Measures
		2.1 Basic Definitions
		2.2 Invariant Measures Arising from Positive Eigenfunctions
		2.3 All Invariant Measures Arise from Positive Eigenfunctions
		2.4 Open Problem
		2.5 Notes and References
	3 Ergodic Properties of Horocycle Invariant Radon Measures
		3.1 Infinite Ergodic Theory
		3.2 Example: mathbbZd-covers of Compact Hyperbolic Surfaces
		3.3 Open Problem
		3.4 Notes and References
	4 Generic Points and Equidistribution
		4.1 Generic Points
		4.2 Sketch of Proof for the Volume Measure
		4.3 Open Problems
		4.4 Notes and References
	5 Proof of the Measure Classification Theorem
		5.1 It Is Enough to Prove g-quasi-invariance
		5.2 The Possible Values of Hµ and the Support of e.i.r.m.
		5.3 Holonomies
		5.4 The Equation R(cdot,eiθ)approxt
		5.5 Proof of the Holonomy Lemmas
		5.6 Proof of the Holonomy Lemmas
		5.7 Summary
		5.8 Notes and References
	6 Appendix 1: Busemann\'s Function
	7 Appendix 2: The Cocycle Reduction Theorem
		7.1 Preliminaries on Countable Equivalence Relations
		7.2 The Cocycle Reduction Theorem
		7.3 The Proof in Case There Are No Square Holes
		7.4 The Proof in Case There Is a Square Hole
		7.5 Notes and References
	References
Higher Order Correlations for Group Actions
	1 Introduction
	2 Decay of Matrix Coefficients
	3 Application: Counting Lattice Points
	4 Quantitative Estimates on Matrix Coefficients
	5 Bounds on Higher Order Correlations
	6 Application: Existence of Configurations
	7 Application: Central Limit Theorem
	References
Exponential Mixing: Lectures from Mumbai
	1 Introduction
		1.1 The Notion of Mixing
		1.2 Definition of Mixing
	2 Discrete Case
		2.1 One-Dimensional Expanding Maps
		2.2 Invariant Measures
	3 The Transfer Operator
		3.1 The Transfer Operator Used to Give Invariant Measures
	4 Continuous Case
		4.1 Geodesic Flows
		4.2 Exponential Decay for Geodesic Flows
	5 Exponential Decay of Correlations for Flows
		5.1 Overview
		5.2 The Simplified Model: Suspension Semi-flow
		5.3 A Short Sketch of the Proof
	6 Other Applications of Transfer Operators
		6.1 Problems that Count (Closed Geodesics)
		6.2 Multiple Mixing for Geodesic Flows
		6.3 Skew Products
		6.4 Skew Products and Flows
		6.5 Euclidean Algorithm
	References