
Rigidity in Dynamics and Geometry
Contributions from the Programme Ergodic Theory, Geometric Rigidity and Number Theory, Isaac Newton Institute for the Mathematical Sciences Cambridge, United Kingdom, 5 January - 7 July 2000
By: Marc Burger (Editor), Alessandra Iozzi (Editor)
Hardcover | 13 May 2002
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512 Pages
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| Quasi-Conformal Geometry and Hyperbolic Geometry | p. 1 |
| Introduction | p. 1 |
| Basic Tools in Geometric Function Theory and Quasi-Conformal Geometry | p. 1 |
| Gromov Hyperbolic Groups: a Motivation for Quasi-Conformal Geometry | p. 5 |
| Further Properties of Loewner Spaces and Spaces with Poincaré Inequalities | p. 8 |
| Differentiability of Quasi-Symmetric Homeomorphisms, Applications to Quasi-Isometry | p. 11 |
| References | p. 15 |
| On and Around the Bounded Cohomology of SL2 | p. 19 |
| Introduction | p. 19 |
| Notations and Conventions | p. 22 |
| A Differential Group | p. 23 |
| Constructing Cocycles | p. 29 |
| Above Degree Two | p. 33 |
| References | p. 36 |
| Densite d'orbites d'actions de groupes lineaires et proprietes d'equidistribution de marches aleatoires | p. 39 |
| Introduction | p. 39 |
| Ensemble limite, ensemble asymptotique | p. 41 |
| Densite d'orbites de dans L(Rd) et équirépartition | p. 47 |
| Actions sur les varietes de drapeaux homogènes | p. 59 |
| La methode des equations fonctionnelles | p. 62 |
| Actions des sous-groupes N et A sur G et G/M | p. 68 |
| Références | p. 74 |
| Exceptional Sets in Dynamical Systems and Diophantine Approximation | p. 77 |
| Introduction | p. 77 |
| Rotation Number | p. 78 |
| Very Well Approximable Numbers and Khintchine's Theorem | p. 82 |
| Kolmogorov-Arnol'd-Moser (KAM) Theory | p. 84 |
| Linearisation | p. 87 |
| Diophantine Approximation in Hyperbolic Geometry | p. 89 |
| Extremal Manifolds and Flows | p. 93 |
| Conclusion | p. 94 |
| References | p. 95 |
| An Introduction to Cocycle Super-Rigidity | p. 99 |
| Introduction | p. 99 |
| Cocycles over Group Actions | p. 100 |
| Cocycles and Principal Bundle Actions | p. 105 |
| Semisimple Lie Groups, in a Hurry | p. 113 |
| All the Ergodic Theory We Need | p. 117 |
| The Algebraic Hull | p. 120 |
| Super-Rigidity | p. 123 |
| The Proof | p. 124 |
| A Few Immediate Applications | p. 131 |
| References133 | |
| Rigid Geometric Structures and Representations of Fundamental Groups | p. 135 |
| Introduction | p. 135 |
| Rigid Structures, Killing Fields and Representations of Fundamental Groups | p. 137 |
| Representations of the Fundamental Group and Dynamics of the Action | p. 139 |
| Constructing Quotients from Representations of 1 | p. 143 |
| References | p. 146 |
| Coarse-Geometric Perspective on Negatively Curved Manifolds and Groups | p. 149 |
| Introduction | p. 149 |
| The Geometric Setup | p. 150 |
| Basic Notions of the Coarse-Geometric Setup | p. 155 |
| Some Results | p. 159 |
| References | p. 165 |
| On Orbit Equivalence of Measure Preserving Actions | p. 167 |
| Equivalence Relations | p. 167 |
| Measure Equivalence | p. 170 |
| Cost of an Equivalence Relation | p. 174 |
| l2 Betti Numbers for Groups | p. 177 |
| Simplicial Actions of an Equivalence Relation | p. 180 |
| Actions of the Equivalence Relation on a Simplicial Complex | p. 182 |
| l2 Betti Numbers for Equivalence Relations and Their Actions | p. 183 |
| References | p. 184 |
| The Margulis Invariant of Isometric Actions on Minkowski (2+1)-Space | p. 187 |
| Introduction | p. 187 |
| Affino Representations | p. 188 |
| Lorentzian Geometry | p. 190 |
| Deformation Theory | p. 191 |
| Properness | p. 194 |
| Linear Growth | p. 195 |
| Triangle Group Deformations | p. 196 |
| References | p. 198 |
| Diophantine Approximation in Negatively Curved Manifolds and in the Heisenberg Group | p. 203 |
| Introduction | p. 203 |
| The Survey Part | p. 204 |
| Diophantine Approximation in the Heisenberg Group | p. 213 |
| References | p. 225 |
| Appendix: Diophantine Approximation on Hyperbolic Surfaces | p. 227 |
| References | p. 236 |
| Bounded Cohomology, Boundary Maps, and Rigidity of Representations into Homeo+(S1) and SU(1,n) | p. 237 |
| Introduction | p. 237 |
| The Euler Class and the Orientation Cocyclo | p. 242 |
| The Proof of the "Formula" | p. 243 |
| $$-Equivariant Measurable Maps into M(S1) | p. 248 |
| Semiconjugacy and the Proofs of Matsumoto's and Goldman's Theorems | p. 251 |
| References | p. 259 |
| SAT Actions and Ergodic Properties of the Horosphere Foliation | p. 261 |
| Ergodic Properties of SAT Actions | p. 261 |
| The Horosphere Foliation and the Horocycle Flow | p. 265 |
| Ergodicity of Busemann Cocycles | p. 269 |
| References | p. 280 |
| Nonexpanding Maps, Busemann Functions, and Multiplicative Ergodic Theory | p. 283 |
| Introduction | p. 283 |
| Busemann Functions | p. 284 |
| Subadditivity | p. 285 |
| Nonexpanding Maps with Unbounded Orbit | p. 286 |
| Multiplicative Ergodic Theory | p. 288 |
| Nonexpansive Iterates in Banach Spaces | p. 291 |
| References | p. 292 |
| The Phase Space of fc-Surfaces | p. 295 |
| Presentation | p. 295 |
| The Geodesic Flow | p. 295 |
| One More Dimension | p. 296 |
| A Hyperbolic Example | p. 298 |
| Geometric Properties of fe-Surfaces and Examples | p. 298 |
| Phase Space | p. 300 |
| Transversal Measure and Coding | p. 302 |
| Questions | p. 305 |
| References | p. 307 |
| Schottky Subgroups of Mapping Class Groups and the Geometry of Surface-by-Free Groups | p. 309 |
| Introduction | p. 309 |
| Schottky Subgroups of Mapping Class Groups | p. 310 |
| Geometry of Surface-by-Schottky Groups | p. 313 |
| Stable Quasi-Geodesics in Teichmuller Space and the Ending Lamination Conjecture | p. 316 |
| References | p. 318 |
| Actions of Semisimple Lie Groups with Stationary Measure | p. 321 |
| Introduction | p. 321 |
| Examples of Actions Without an Invariant Measure | p. 322 |
| Stationary Measures | p. 323 |
| A Structure Theorem for Stationary Measures | p. 326 |
| Real Rank One Groups: Some Constructions | p. 328 |
| Structure Theorems: Groups of Real Rank at Least Two | p. 330 |
| Ergodicity Conditions and the Existence of Projective Factors of Full Entropy | p. 333 |
| Construction of Projective Factors or Actions of Factor Groups | p. 338 |
| Expanding Versus Contracting Automorphisms, and Margulis' Normal Subgroup Theorem | p. 341 |
| References | p. 342 |
| On the Cohomology of Anosov Actions | p. 345 |
| Cocycles | p. 345 |
| Partially Hyperbolic and Anosov Actions | p. 347 |
| Livsic Theory | p. 348 |
| Regularity Results | p. 351 |
| Higher Rank Abelian Actions | p. 353 |
| Applications to the Rigidity of Higher Rank Lattice Actions | p. 356 |
| References | p. 359 |
| Harmonic Analysis and Hecke Operators | p. 363 |
| Uniform Pointwise Bounds s for Matrix Coefficients | p. 363 |
| Equidistribution of Hecke Points | p. 367 |
| Equidistribution of Integer Points on a Family of Homogeneous Varieties | p. 370 |
| Distributing Points on the Spheres Sn (n ≥ 4) | p. 374 |
| References | p. 376 |
| Lp-Cohomology and Pinching | p. 379 |
| Lp-Cohomology | p. 381 |
| The Kürmeth Formula | p. 383 |
| Pinched Manifolds | p. 386 |
| Non-Vanishing of Torsion | p. 386 |
| Vanishing of Torsion for H2c | p. 388 |
| References | p. 388 |
| Classical and Non-Linearity Properties of Kac-Moody Lattices | p. 391 |
| Introduction | p. 391 |
| A Classical Arithmetic Situation and its Geometric Formulation | p. 392 |
| Kac Moody Theory and the Generalization | p. 395 |
| Questions Arising from the Generalization | p. 400 |
| References | p. 404 |
| Actions of Maximal Tori on Homogeneous spaces | p. 407 |
| Introduction | p. 407 |
| if-Algebraic Groups, Arithmetic Subgroups and Central Simple Algebras | p. 412 |
| Reductive Q-Subgroups of SL1 (A) | p. 416 |
| Homogeneous Orbit Closures Containing Relatively Compact T-Orbits | p. 419 |
| Closed T-Orbits | p. 421 |
| References | p. 423 |
| Dynamics on Parameter Spaces: Submanifold and Fractal Subset Questions | p. 425 |
| Introduction | p. 425 |
| Conjectures and Results | p. 427 |
| A Gentle Reminder Regarding Dynamics on Homogeneous / Quadratic Differential Spaces | p. 429 |
| Quantitative Nondivergence and Applications | p. 432 |
| Khinchin's Convergence Case for Fractals | p. 434 |
| Logarithm Laws on a Teichmüller Horocycle | p. 437 |
| References | p. 439 |
| Superrigid Subgroups and Syndetic Hulls in Solvable Lie Groups | p. 441 |
| What Is a Superrigid Subgroup? | p. 441 |
| Other Superrigidity Theorems | p. 445 |
| Our Prototypical Proof of Superrigidity | p. 448 |
| Solvable Lie Groups and Zariski Closed Subgroups | p. 452 |
| Existence of Syndetic Hulls | p. 454 |
| References | p. 457 |
| Square Tiled Surfaces and Teichmüller Volumes of the Moduli Spaces of Abelian Differential | p. 459 |
| Motivations | p. 459 |
| Translation Surfaces Versus Flat Surfaces | p. 460 |
| Moduli Spaces of Abelian Differentials | p. 461 |
| Counting Volume by Means of Counting Integer Points | p. 462 |
| Two Examples of Computation | p. 463 |
| Volumes of Some Strata of Abelian Differentials | p. 467 |
| Lyapunov Exponents of the Teichmüller Geodesic Flow | p. 467 |
| Conjectural Probability P(n) of n Bands of Trajectories for a Rational Interval Exchange Transformation | p. 469 |
| References | p. 471 |
| On Property (T) for Discrete Groups | p. 473 |
| Introduction | p. 473 |
| Expanders | p. 474 |
| How to Prove Property (T) | p. 476 |
| Random Groups | p. 478 |
| References | p. 480 |
| Index | p. 483 |
| Table of Contents provided by Publisher. All Rights Reserved. |
ISBN: 9783540432432
ISBN-10: 3540432434
Published: 13th May 2002
Format: Hardcover
Language: English
Number of Pages: 512
Audience: College, Tertiary and University
Publisher: Springer Nature B.V.
Country of Publication: DE
Dimensions (cm): 23.5 x 16.51 x 3.81
Weight (kg): 0.89
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