Contents Preface vii Acknowledgments ix Lecture 1. Introduction 1 1A. Discrete versions of G 1 1B. Nonlinear actions 2 1C. Open questions 5 Comments 5 References 6 Lecture 2. Actions in Dimension 1 or 2 9 2A. Finite actions 9 2B. Actions on the circle 10 2C. Farb-Shalen method 11 2D. Actions on surfaces 12 Comments 13 References 14 Lecture 3. Geometric Structures 17 3A. Reductions of principal bundles 17 3B. Algebraic hull of a G-action on M 18 3C. Actions preserving an H-structure 20 3D. Groups that act on Lorentz manifolds 21 Comments 22 References 22 Lecture 4. Fundamental Groups I 25 4A. Engaging conditions 25 4B. Consequences 27 4C. Examples: Actions with engaging conditions 29 Comments 30 References 30 Lecture 5. Gromov Representation 33 5A. Gromov’s Centralizer Theorem 33 5B. Higher-order frames and connections 35 5C. Ideas in the proof of Gromov’s Centralizer Theorem 37 Comments 38 References 38 v

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