
Lectures on Kähler Groups
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This book gives an authoritative and up-to-date introduction to the study of fundamental groups of compact Kähler manifolds, known as Kähler groups. Approaching the subject from the perspective of a geometric group theorist, Pierre Py equips readers with the necessary background in both geometric group theory and Kähler geometry, covering topics such as the actions of Kähler groups on spaces of nonpositive curvature, the large-scale geometry of infinite covering spaces of compact Kähler manifolds, and the topology of level sets of pluriharmonic functions.
Presenting the most important results from the past three decades, the book provides graduate students and researchers with detailed original proofs of several central theorems, including Gromov and Schoen's description of Kähler group actions on trees; the study of solvable quotients of Kähler groups following the works of Arapura, Beauville, Campana, Delzant, and Nori; and Napier and Ramachandran's work characterizing covering spaces of compact Kähler manifolds having many ends. It also describes without proof many of the recent breakthroughs in the field.
Lectures on Kähler Groups also gives, in eight appendixes, detailed introductions to such topics as the study of ends of groups and spaces, groups acting on trees and Hilbert spaces, potential theory, and L2 cohomology on Riemannian manifolds.
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Content
- Cover
- Contents
- Preface
- 1. Introduction
- 1.1 Kähler Groups
- 1.2 What Is to Be Found in This Book?
- 1.3 What Is Not to Be Found in This Book?
- 1.4 Problems
- 1.5 Some Conventions
- 2. Riemann Surfaces and Orbifolds
- 2.1 Orbifold Structures on Surfaces
- 2.2 The Orbifold Structure Induced by a Fibration
- 2.3 A Finiteness Result
- 2.4 Recognizing Fibrations
- 3. The Fibration Problem: From Infinite to Finite Covers
- 3.1 From Infinite to Finite Covers
- 3.2 From a Compact Leaf to a Fibration
- 4. The Theorem of Castelnuovo-de Franchis and Its Variants
- 4.1 The Classical Castelnuovo-de Franchis Theorem
- 4.2 Castelnuovo-de Franchis and Flat Bundles
- 4.3 The L2 Case
- 4.4 Gromov's Cup Product Lemma and Its Variants
- 5. Many Fibration Criteria
- 5.1 A Fibration Criterion for Proper Pluriharmonic Maps
- 5.2 The Use of Plurisubharmonic Functions
- 5.3 Totaro and Pereira's Criterion
- 6. Kähler Groups and Trees
- 6.1 Actions on Trees and Harmonic Functions
- 6.2 The Kähler Case
- 6.3 Applications
- 7. Covering Spaces of Compact Kähler Manifolds and Ends
- 7.1 Ends and Kähler Groups: Some Motivation
- 7.2 The 3-Ends Theorem: The Non-amenable Case
- 7.3 The 3-Ends Theorem: The General Case
- 7.4 The Case of Filtered Ends
- 8. Representations into PSL2(C)
- 8.1 Bass's Theorem
- 8.2 Arithmetic Lattices in PSL2(R)l: A Short Account
- 8.3 Representations of Kähler Groups into PSL2(C)
- 9. Harmonic Maps to Locally Symmetric Spaces
- 9.1 Harmonic Maps on Riemannian Manifolds
- 9.2 A Brief Introduction to Nonabelian Hodge Theory
- 9.3 Real Hyperbolic Spaces
- 9.4 Complex Hyperbolic Spaces
- 10. Lattices and Groups of Hodge Type
- 10.1 Lie Groups of Hodge Type
- 10.2 Simpson's Theorem
- 10.3 The Link with Hodge Theory
- 11. The Bieri-Neumann-Strebel Invariant
- 11.1 The BNS Invariant of a Finitely Generated Group
- 11.2 Some Results on the BNS Invariant
- 11.3 Delzant's Theorem
- 11.4 Some Applications
- 12. The Green-Lazarsfeld Set
- 12.1 The Green-Lazarsfeld Set of a Finitely Generated Group
- 12.2 Preliminaries on Flat Line Bundles
- 12.3 Green-Lazarsfeld and Bieri-Neumann-Strebel
- 12.4 Solvable Quotients of Kähler Groups
- 13. Actions on Real Trees
- 13.1 Kernels of Hyperbolic Type and Trees
- 13.2 A Factorization Result for Actions on Real Trees
- 13.3 Final Comments
- Appendices
- A. Ends of Spaces and Groups
- A.1 The Space of Ends
- A.2 Relative Ends
- A.3 Filtered Ends
- B. Groups Acting on Trees
- B.1 Preliminaries
- B.2 Free Groups in Groups Acting on Trees
- B.3 Trees and Cocycles
- B.4 The Tree of GL2
- B.5 Real Trees
- B.6 Closed 1-Forms and Real Trees
- C. Affine Actions on Hilbert Spaces
- C.1 Orthogonal Representations and Affine Actions
- C.2 Harmonic Maps and Stability
- C.3 Stability, Non-amenability, and Schreier Graphs
- C.4 The First l2-Betti Number
- D. Harmonic Functions of Finite Energy
- D.1 Construction of Harmonic Functions with Prescribed Behavior on Ends
- D.2 L2-Cohomology and Cohomology with Values in a Unitary Representation
- D.3 Pluriharmonicity for Finite Energy Functions
- E. Potential Theory
- E.1 Hyperbolic and Parabolic Riemannian Manifolds
- E.2 Harmonic Functions on Riemannian Manifolds All of Whose Ends Are Hyperbolic
- E.3 The Parabolic Case
- F. Nakai's Theorem
- F.1 Green Function and Stone-Cech Compactification
- F.2 Transfinite Diameter, Tchebycheff Constant, and Capacity
- F.3 Construction of a Proper Harmonic Exhaustion
- G. Diederich and Mazzilli's Theorem
- G.1 A Uniform Volume Estimate
- G.2 The Globalization Result
- G.3 Proof of Diederich and Mazzilli's Theorem
- H. Harmonic Maps and Nonpositive Hermitian Curvature
- H.1 An Identity Involving Hermitian Curvature
- H.2 Curvature Calculations
- Bibliography
- Index
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