Geometry, Topology, and Dynamics in Negative Curvature (London Mathematical Society Lecture Note Series, Series Number 425)

The Icm 2010 Satellite Conference 'geometry, Topology And Dynamics In Negative Curvature' Afforded An Excellent Opportunity To Discuss Various Aspects Of This Fascinating Interdisciplinary Subject In Which Methods And Techniques From Geometry, Topology, And Dynamics Often Interact In Novel And Interesting Ways. Containing Ten Survey Articles Written By Some Of The Leading Experts In The Field, This Proceedings Volume Provides An Overview Of Important Recent Developments Relating To Negative Curvature. Topics Covered Include Homogeneous Dynamics, Harmonic Manifolds, The Atiyah Conjecture, Counting Circles And Arcs, And Hyperbolic Buildings. Each Author Pays Particular Attention To The Expository Aspects, Making The Book Particularly Useful For Graduate Students And Mathematicians Interested In Transitioning From Other Areas Via The Common Theme Of Negative Curvature--back Cover. Machine Generated Contents Note: 1.gap Distributions And Homogeneous Dynamics / Jayadev S. Athreya -- 1.introduction -- 2.meta-theorems On Gap Distributions -- 3.farey Sequences -- 4.saddle Connections -- 5.abstract Homogeneous Dynamics -- 6.further Questions -- References -- 2.topology Of Open Nonpositively Curved Manifolds / Igor Belegradek -- 1.flavors Of Negative Curvature -- 2.manifolds Of Dimensions Two And Three -- 3.towards A Rough Classification Of Discrete Isometry Groups -- 4.groups Of Non-parabolic Isometries -- 5.groups With Rank One Elements: Prelude -- 6.acylindrically Hyperbolic Groups And Rank One Elements -- 7.bounded Cohomology And Rank One Elements -- 8.monod-shalom's Class And Rank One Elements -- 9.groups That Fix A Point At Infinity: Prelude -- 10.groups Whose Center Contains A Parabolic -- 11.anchored Groups And Fixed Points At Infinity -- 12.homotopy Obstructions (after Gromov And Izeki-nayatani) -- 13.homeomorphism Obstructions: Exploiting R Factors -- 14.benefits Of A Lower Curvature Bound -- 15.injectivity Radius Going To Zero At Infinity -- 16.negatively Curved Manifolds With Uniform Volume Bound -- 17.non-aspherical Ends Of Nonpositively Curved Manifolds -- 18.riemannian Hyperbolization (after Ontaneda) -- 19.topology Of Known Manifolds Of Bounded Negative Curvature -- 20.zoo Of Finite Volume Rank One Manifolds -- References -- 3.cohomologie Et Actions Isometriques Propres Sur Les Espaces Lp / Marc Bourdon -- 1.introduction -- 2.cohomologie Lp -- 3.actions Isometriques Propres Sur Les Espaces Lp -- References -- 4.compact Clifford -- Klein Forms -- Geometry, Topology And Dynamics / David Constantine -- 1.introduction -- 2.notation -- 3.topology -- 4.geometry -- 5.dynamics -- 6.deformations And Moduli Spaces Of Compact Forms -- 7.the Road Ahead -- References -- 5.a Survey On Noncompact Harmonic And Asymptotically Harmonic Manifolds / Gerhard Knieper -- 1. Introduction -- 2.basics On Jacobi Tensors On Manifolds Without Conjugate Points -- 3.volume Growth On Asymptotically Harmonic Manifolds -- 4.density Functions Of Harmonic Manifolds -- 5.harmonic Manifolds With Subexponential Volume Growth Are Flat -- 6.the Rank Of An Asymptotically Harmonic Manifold -- 7.characterization Of Hyperbolicity For Asymptotically Harmonic Manifolds -- 8.harmonic Manifolds With Bounded Asymptote -- 9.harmonic And Asymptotically Harmonic Manifolds Without Focal Points -- References -- 6.the Atiyah Conjecture / Peter A. Linnell -- 1.introduction -- 2.notation And Terminology -- 3.group Von Neumann Algebras And Dimension -- 4.formulation Of The Atiyah Conjecture -- 5.early Results On The Atiyah Conjecture -- 6.approximation Techniques -- 7.finite Extensions -- 8.pro-p Groups Of Finite Rank -- 9.approximation Over Arbitrary Fields -- References -- 7.cannon-thurston Maps For Surface Groups: An Exposition Of Amalgamation Geometry And Split Geometry / Mahan Mj -- 1.introduction -- 2.preliminaries And Amalgamation Geometry -- 3.relative Hyperbolicity -- 4.universal Covers Of Building Blocks And Electric Geometry -- 5.construction Of Quasiconvex Sets And Quasigeodesics -- 6.cannon-thurston Maps For Surfaces Without Punctures -- 7.weakening The Hypothesis I: Graph Quasiconvexity And Graph Amalgamation Geometry -- 8.weakening The Hypothesis Ii: Split Geometry -- References -- 8.counting Visible Circles On The Sphere And Kleinian Groups / Nimish Shah -- 1.introduction -- 2.equidistribution Of Normal Translates Of A Hyperbolic Surface -- 3.proof Of Theorem 3 -- 4.counting With Respect To Euclidean Curvature -- References -- 9.counting Arcs In Negative Curvature / Frederic Paulin -- 1.introduction -- 2.geometry And Dynamics In Negative Curvature -- 3.common Perpendiculars Of Convex Sets -- 4.using Eskin-mcmullen's Equidistribution Theorem -- 5.arithmetic Applications -- 6.patterson, Bowen-margulis And Skinning Measures -- 7.finite Volume Hyperbolic Manifolds -- 8.the Main Counting Result Of Common Perpendiculars -- 9.spectral Gaps, Exponential Decay Of Correlations And Error Terms -- 10.gibbs Measures And Counting Arcs With Weights -- References -- 10.lattices In Hyperbolic Buildings / Anne Thomas -- 1.buildings And Hyperbolic Buildings -- 2.lattices. Edited By C.s. Aravinda, Tifr Centre For Applicable Mathematics, Bangalore, F.t. Farrell, Tsinghua University, Beijing, J.-f. Lafont, Ohio State University, Columbus. Includes Bibliographical References.