Cover

Series page

Title

Copyright

Contents

List of contributors

Preface

1 Gap distributions and homogeneous dynamics

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

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 uniformvolume 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 isométriques propres sur les espaces L(sub[p])

1 Introduction

2 Cohomologie l(sub[p])

3 Actions isometriques propres sur les espaces L(sub[p])

References

4 Compact Clifford–Klein Forms – Geometry, Topology and Dynamics

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

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

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

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

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

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

1 Buildings and hyperbolic buildings

2 Lattices

References