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)

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Hardcover | 13 May 2002

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This volume of proceedings is an offspring of the special semester Ergodic Theory, Geometric Rigidity and Number Theory which was held at the Isaac Newton Institute for Mathematical Sciences in Cambridge, UK, from Jan uary until July, 2000. Beside the activities during the semester, there were workshops held in January, March and July, the first being of introductory nature with five short courses delivered over a week. Although the quality of the workshops was excellent throughout the semester, the idea of these proceedings came about during the March workshop, which is hence more prominently represented, The format of the volume has undergone many changes, but what has remained untouched is the enthusiasm of the contributors since the onset of the project: suffice it to say that even though only two months elapsed between the time we contacted the potential authors and the deadline to submit the papers, the deadline was respected in the vast majority of the cases. The scope of the papers is not completely uniform throughout the volume, although there are some points in common. We asked the authors to write papers keeping in mind the idea that they should be accessible to students. At the same time, we wanted the papers not to be a summary of results that appeared somewhere else.

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Non-Fiction Mathematics Calculus & Mathematical Analysis Functional Analysis & Transforms Differential Calculus & Equations Number Theory Geometry Differential & Riemannian Geometry Reference, Information & Interdisciplinary Subjects Research & Information Information theory Cybernetics & Systems Theory

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(R^d) 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
L^p-Cohomology and Pinching	p. 379
L^p-Cohomology	p. 381
The Kürmeth Formula	p. 383
Pinched Manifolds	p. 386
Non-Vanishing of Torsion	p. 386
Vanishing of Torsion for H²_c	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.

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

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