The study of group actions is more than a hundred years old but remains to this day a vibrant and widely studied topic in a variety of mathematic fields. A central development in the last fifty years is the phenomenon of rigidity, whereby one can classify actions of certain groups, such as lattices in semi-simple Lie groups. This provides a way to classify all possible symmetries of important spaces and all spaces admitting given symmetries. Paradigmatic results can be found in the seminal work of George Mostow, Gergory Margulis, and Robert J. Zimmer, among others.
The papers in Geometry, Rigidity, and Group Actions explore the role of group actions and rigidity in several areas of mathematics, including ergodic theory, dynamics, geometry, topology, and the algebraic properties of representation varieties. In some cases, the dynamics of the possible group actions are the principal focus of inquiry. In other cases, the dynamics of group actions are a tool for proving theorems about algebra, geometry, or topology. This volume contains surveys of some of the main directions in the field, as well as research articles on topics of current interest.
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ix | |
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PART 1 Group Actions on Manifolds |
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1 An Extension Criterion for Lattice Actions on the Circle |
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3 | (29) |
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2 Meromorphic Almost Rigid Geometric Structures |
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32 | (27) |
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3 Harmonic Functions over Group Actions |
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59 | (13) |
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4 Groups Acting on Manifolds: Around the Zimmer Program |
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72 | (86) |
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5 Can Lattices in SL (n, R) Act on the Circle? |
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158 | (50) |
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6 Some Remarks on Area-Preserving Actions of Lattices |
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208 | (21) |
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7 Isometric Actions of Simple Groups and Transverse Structures: The Integrable Normal Case |
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229 | (33) |
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8 Some Remarks Inspired by the C0 Zimmer Program |
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262 | (23) |
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PART 2 Analytic, Ergodic, and Measurable Group Theory |
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9 Calculus on Nilpotent Lie Groups |
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285 | (11) |
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10 A Survey of Measured Group Theory |
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296 | (79) |
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11 On Relative Property (T) |
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375 | (21) |
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12 Noncommutative Ergodic Theorems |
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396 | (23) |
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13 Cocycle and Orbit Superrigidity for Lattices in SL (n, R) Acting on Homogeneous Spaces |
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419 | (36) |
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PART 3 Geometric Group Theory |
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14 Heights on SL2 and Free Subgroups |
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455 | (39) |
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15 Displacing Representations and Orbit Maps |
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494 | (21) |
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16 Problems on Automorphism Groups of Nonpositively Curved Polyhedral Complexes and Their Lattices |
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515 | (46) |
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17 The Geometry of Twisted Conjugacy Classes in Wreath Products |
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561 | (30) |
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PART 4 Group Actions on Representations Varieties |
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18 Ergodicity of Mapping Class Group Actions on SU(2)-Character Varieties |
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591 | (18) |
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19 Dynamics of Aut (Fn) Actions on Group Presentations and Representations |
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609 | (36) |
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| List of Contributors |
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645 | |
Benson Farb is professor of mathematics at the University of Chicago. He is the author of Problems on Mapping Class Groups and Related Topics and coauthor of Noncommutative Algebra. David Fisher is professor of mathematics at Indiana University.
