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El. knyga: Geometry and Dynamics in Gromov Hyperbolic Metric Spaces

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While working to expose a coherent general theory of groups acting isometrically on infinite-dimensional hyperbolic space, Das, Simmons,and Urbanski realized that a more natural domain for their inquiries was the much larger setting of semigroups acting on Gromov hyperbolic metric spaces. That way, while answering their own questions about infinite-dimensional hyperspace, they could simultaneously construct a theoretical framework for mathematicians who are interested in more exotic spaces such as the curve graph, arc graph, and arc complex and the free splitting and free factor complexes. Annotation ©2017 Ringgold, Inc., Portland, OR (protoview.com)
List of Figures xi
Prologue xiii
Chapter 1 Introduction and Overview
xvii
1.1 Preliminaries
xviii
1.1.1 Algebraic hyperbolic spaces
xviii
1.1.2 Gromov hyperbolic metric spaces
xviii
1.1.3 Discreteness
xxi
1.1.4 The classification of semigroups
xxii
1.1.5 Limit sets
xxiii
1.2 The Bishop-Jones theorem and its generalization
xxiv
1.2.1 The modified Poincare exponent
xxvii
1.3 Examples
xxviii
1.3.1 Schottky products
xxviii
1.3.2 Parabolic groups
xxix
1.3.3 Geometrically finite and convex-cobounded groups
xxix
1.3.4 Counterexamples
xxx
1.3.5 R-trees and their isometry groups
xxxi
1.4 Patterson-Sullivan theory
xxxi
1.4.1 Quasiconformal measures of geometrically finite groups
xxxiv
1.5 Appendices
xxxv
Part 1 Preliminaries 1(106)
Chapter 2 Algebraic hyperbolic spaces
3(16)
2.1 The definition
3(1)
2.2 The hyperboloid model
4(3)
2.3 Isometries of algebraic hyperbolic spaces
7(5)
2.4 Totally geodesic subsets of algebraic hyperbolic spaces
12(3)
2.5 Other models of hyperbolic geometry
15(4)
2.5.1 The (Klein) ball model
16(1)
2.5.2 The half-space model
16(2)
2.5.3 Transitivity of the action of Isom(H) on partial differential H
18(1)
Chapter 3 R-trees, CAT(-1) spaces, and Gromov hyperbolic metric spaces
19(30)
3.1 Graphs and R-trees
19(3)
3.2 CAT(-1) spaces
22(2)
3.2.1 Examples of CAT(-1) spaces
23(1)
3.3 Gromov hyperbolic metric spaces
24(3)
3.3.1 Examples of Gromov hyperbolic metric spaces
26(1)
3.4 The boundary of a hyperbolic metric space
27(8)
3.4.1 Extending the Gromov product to the boundary
29(3)
3.4.2 A topology on bord X
32(3)
3.5 The Gromov product in algebraic hyperbolic spaces
35(5)
3.5.1 The Gromov boundary of an algebraic hyperbolic space
39(1)
3.6 Metrics and metametrics on bord X
40(9)
3.6.1 General theory of metametrics
40(2)
3.6.2 The visual metametric based at a point w belongs to X
42(1)
3.6.3 The extended visual metric on bord X
43(2)
3.6.4 The visual metametric based at a point xi belongs to partialdifferential X
45(4)
Chapter 4 More about the geometry of hyperbolic metric spaces
49(20)
4.1 Gromov triples
49(1)
4.2 Derivatives
50(4)
4.2.1 Derivatives of metametrics
50(1)
4.2.2 Derivatives of maps
51(2)
4.2.3 The dynamical derivative
53(1)
4.3 The Rips condition
54(1)
4.4 Geodesics in CAT(-1) spaces
55(6)
4.5 The geometry of shadows
61(5)
4.5.1 Shadows in regularly geodesic hyperbolic metric spaces
61(1)
4.5.2 Shadows in hyperbolic metric spaces
61(5)
4.6 Generalized polar coordinates
66(3)
Chapter 5 Discreteness
69(10)
5.1 Topologies on Isom(X)
69(3)
5.2 Discrete groups of isometries
72(7)
5.2.1 Topological discreteness
74(2)
5.2.2 Equivalence in finite dimensions
76(1)
5.2.3 Proper discontinuity
76(2)
5.2.4 Behavior with respect to restrictions
78(1)
5.2.5 Countability of discrete groups
78(1)
Chapter 6 Classification of isometries and semigroups
79(12)
6.1 Classification of isometries
79(3)
6.1.1 More on loxodromic isometries
81(1)
6.1.2 The story for real hyperbolic spaces
82(1)
6.2 Classification of semigroups
82(3)
6.2.1 Elliptic semigroups
83(1)
6.2.2 Parabolic semigroups
83(1)
6.2.3 Loxodromic semigroups
84(1)
6.3 Proof of the Classification Theorem
85(2)
6.4 Discreteness and focal groups
87(4)
Chapter 7 Limit sets
91(16)
7.1 Modes of convergence to the boundary
91(2)
7.2 Limit sets
93(2)
7.3 Cardinality of the limit set
95(1)
7.4 Minimality of the limit set
96(3)
7.5 Convex hulls
99(3)
7.6 Semigroups which act irreducibly on algebraic hyperbolic spaces
102(1)
7.7 Semigroups of compact type
103(4)
Part 2 The Bishop-Jones theorem 107(24)
Chapter 8 The modified Poincare exponent
109(6)
8.1 The Poincare exponent of a semigroup
109(1)
8.2 The modified Poincare exponent of a semigroup
110(5)
Chapter 9 Generalization of the Bishop-Jones theorem
115(16)
9.1 Partition structures
116(4)
9.2 A partition structure on partial differential X
120(7)
9.3 Sufficient conditions for Poincare regularity
127(4)
Part 3 Examples 131(86)
Chapter 10 Schottky products
133(16)
10.1 Free products
133(1)
10.2 Schottky products
134(1)
10.3 Strongly separated Schottky products
135(7)
10.4 A partition-structure-like structure
142(4)
10.5 Existence of Schottky products
146(3)
Chapter 11 Parabolic groups
149(16)
11.1 Examples of parabolic groups acting on Einfinity
149(6)
11.1.1 The Haagerup property and the absence of a Margulis lemma
150(1)
11.1.2 Edelstein examples
151(4)
11.2 The Poincare exponent of a finitely generated parabolic group
155(10)
11.2.1 Nilpotent and virtually nilpotent groups
156(1)
11.2.2 A universal lower bound on the Poincare exponent
157(1)
11.2.3 Examples with explicit Poincare exponents
158(7)
Chapter 12 Geometrically finite and convex-cobounded groups
165(24)
12.1 Some geometric shapes
165(3)
12.1.1 Horoballs
165(2)
12.1.2 Dirichlet domains
167(1)
12.2 Cobounded and convex-cobounded groups
168(4)
12.2.1 Characterizations of convex-coboundedness
170(2)
12.2.2 Consequences of convex-coboundedness
172(1)
12.3 Bounded parabolic points
172(4)
12.4 Geometrically finite groups
176(13)
12.4.1 Characterizations of geometrical finiteness
177(5)
12.4.2 Consequences of geometrical finiteness
182(4)
12.4.3 Examples of geometrically finite groups
186(3)
Chapter 13 Counterexamples
189(10)
13.1 Embedding R-trees into real hyperbolic spaces
189(4)
13.2 Strongly discrete groups with infinite Poincare exponent
193(1)
13.3 Moderately discrete groups which are not strongly discrete
193(1)
13.4 Poincare irregular groups
194(4)
13.5 Miscellaneous counterexamples
198(1)
Chapter 14 R-trees and their isometry groups
199(18)
14.1 Construction of IR-trees by the cone method
199(3)
14.2 Graphs with contractible cycles
202(2)
14.3 The nearest-neighbor projection onto a convex set
204(1)
14.4 Constructing IR-trees by the stapling method
205(4)
14.5 Examples of ER-trees constructed using the stapling method
209(8)
Part 4 Patterson-Sullivan theory 217(50)
Chapter 15 Conformal and quasiconformal measures
219(10)
15.1 The definition
219(1)
15.2 Conformal measures
220(1)
15.3 Ergodic decomposition
220(2)
15.4 Quasiconformal measures
222(7)
15.4.1 Pointmass quasiconformal measures
223(1)
15.4.2 Non-pointmass quasiconformal measures
224(5)
Chapter 16 Patterson-Sullivan theorem for groups of divergence type
229(12)
16.1 Samuel-Smirnov compactifications
229(1)
16.2 Extending the geometric functions to X
230(2)
16.3 Quasiconformal measures on X
232(2)
16.4 The main argument
234(3)
16.5 End of the argument
237(1)
16.6 Necessity of the generalized divergence type assumption
238(1)
16.7 Orbital counting functions of nonelementary groups
239(2)
Chapter 17 Quasiconformal measures of geometrically finite groups
241(26)
17.1 Sufficient conditions for divergence type
241(3)
17.2 The global measure formula
244(3)
17.3 Proof of the global measure formula
247(6)
17.4 Groups for which µ is doubling
253(6)
17.5 Exact dimensionality of µ
259(8)
17.5.1 Diophantine approximation on ^
261(3)
17.5.2 Examples and non-examples of exact dimensional measures
264(3)
Appendix A. Open problems 267(2)
Appendix B. Index of defined terms 269(6)
Bibliography 275
Tushar Das, University of Wisconsin, La Crosse, WI.

David Simmons, University of York, United Kingdom.

Mariusz Urbanski, University of North Texas, Denton, TX.
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