viii CONTENT S
5. Non-linear groups 85
6. Harmonic maps to trees 87
Chapter 7. Non-Abelian Hodge theory 91
1. Basic concepts 91
2. Yang-Mills equations and the C*-action on Higgs bundles 96
3. Hyperkahler structures and complete integrability 100
4. Applications 103
Chapter 8. Positive results for infinite groups 109
1. Introduction 109
2. The first construction 111
3. A Lefschetz theorem for smooth open varieties 113
4. The general construction 113
5. Non-residually finite Kahler groups 115
Appendix A. Pro group theory 121
1. Definitions of group completions 121
2. Nilpotent completions 124
3. Comparison of nilpotent completions 126
Appendix B. A glossary of Hodge theory 129
Bibliography 133
Index
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