Geometry and Dynamics in Gromov Hyperbolic Metric Spaces: With an Emphasis on Non-proper Settings (Mathematical Surveys and Monographs)

Cover
Title page
Dedication
Contents
List of Figures
Prologue
Chapter 1. Introduction and Overview
	1.1. Preliminaries
		1.1.1. Algebraic hyperbolic spaces
		1.1.2. Gromov hyperbolic metric spaces
		1.1.3. Discreteness
		1.1.4. The classification of semigroups
		1.1.5. Limit sets
	1.2. The Bishop–Jones theorem and its generalization
		1.2.1. The modified Poincaré exponent
	1.3. Examples
		1.3.1. Schottky products
		1.3.2. Parabolic groups
		1.3.3. Geometrically finite and convex-cobounded groups
		1.3.4. Counterexamples
		1.3.5. \\R-trees and their isometry groups
	1.4. Patterson–Sullivan theory
		1.4.1. Quasiconformal measures of geometrically finite groups
	1.5. Appendices
Part  1 .  Preliminaries
	Chapter 2. Algebraic hyperbolic spaces
		2.1. The definition
		2.2. The hyperboloid model
		2.3. Isometries of algebraic hyperbolic spaces
		2.4. Totally geodesic subsets of algebraic hyperbolic spaces
		2.5. Other models of hyperbolic geometry
			2.5.1. The (Klein) ball model
			2.5.2. The half-space model
			2.5.3. Transitivity of the action of \\Isom() on ∂˝
	Chapter 3. \\R-trees, CAT(-1) spaces, and Gromov hyperbolic metric spaces
		3.1. Graphs and \\R-trees
		3.2. CAT(-1) spaces
			3.2.1. Examples of CAT(-1) spaces
		3.3. Gromov hyperbolic metric spaces
			3.3.1. Examples of Gromov hyperbolic metric spaces
		3.4. The boundary of a hyperbolic metric space
			3.4.1. Extending the Gromov product to the boundary
			3.4.2. A topology on \\bord????
		3.5. The Gromov product in algebraic hyperbolic spaces
			3.5.1. The Gromov boundary of an algebraic hyperbolic space
		3.6. Metrics and metametrics on \\bord????
			3.6.1. General theory of metametrics
			3.6.2. The visual metametric based at a point \\notzero∈????
			3.6.3. The extended visual metric on \\bord????
			3.6.4. The visual metametric based at a point ????∈\\del????
	Chapter 4. More about the geometry of hyperbolic metric spaces
		4.1. Gromov triples
		4.2. Derivatives
			4.2.1. Derivatives of metametrics
			4.2.2. Derivatives of maps
			4.2.3. The dynamical derivative
		4.3. The Rips condition
		4.4. Geodesics in CAT(-1) spaces
		4.5. The geometry of shadows
			4.5.1. Shadows in regularly geodesic hyperbolic metric spaces
			4.5.2. Shadows in hyperbolic metric spaces
		4.6. Generalized polar coordinates
	Chapter 5. Discreteness
		5.1. Topologies on \\Isom(????)
		5.2. Discrete groups of isometries
			5.2.1. Topological discreteness
			5.2.2. Equivalence in finite dimensions
			5.2.3. Proper discontinuity
			5.2.4. Behavior with respect to restrictions
			5.2.5. Countability of discrete groups
	Chapter 6. Classification of isometries and semigroups
		6.1. Classification of isometries
			6.1.1. More on loxodromic isometries
			6.1.2. The story for real hyperbolic spaces
		6.2. Classification of semigroups
			6.2.1. Elliptic semigroups
			6.2.2. Parabolic semigroups
			6.2.3. Loxodromic semigroups
		6.3. Proof of the Classification Theorem
		6.4. Discreteness and focal groups
	Chapter 7. Limit sets
		7.1. Modes of convergence to the boundary
		7.2. Limit sets
		7.3. Cardinality of the limit set
		7.4. Minimality of the limit set
		7.5. Convex hulls
		7.6. Semigroups which act irreducibly on algebraic hyperbolic spaces
		7.7. Semigroups of compact type
Part  2 .  The Bishop–Jones theorem
	Chapter 8. The modified Poincaré exponent
		8.1. The Poincaré exponent of a semigroup
		8.2. The modified Poincaré exponent of a semigroup
	Chapter 9. Generalization of the Bishop–Jones theorem
		9.1. Partition structures
		9.2. A partition structure on \\del????
		9.3. Sufficient conditions for Poincaré regularity
Part  3 .  Examples
	Chapter 10. Schottky products
		10.1. Free products
		10.2. Schottky products
		10.3. Strongly separated Schottky products
		10.4. A partition-structure–like structure
		10.5. Existence of Schottky products
	Chapter 11. Parabolic groups
		11.1. Examples of parabolic groups acting on \\E^{∞}
			11.1.1. The Haagerup property and the absence of a Margulis lemma
			11.1.2. Edelstein examples
		11.2. The Poincaré exponent of a finitely generated parabolic group
			11.2.1. Nilpotent and virtually nilpotent groups
			11.2.2. A universal lower bound on the Poincaré exponent
			11.2.3. Examples with explicit Poincaré exponents
	Chapter 12. Geometrically finite and convex-cobounded groups
		12.1. Some geometric shapes
			12.1.1. Horoballs
			12.1.2. Dirichlet domains
		12.2. Cobounded and convex-cobounded groups
			12.2.1. Characterizations of convex-coboundedness
			12.2.2. Consequences of convex-coboundedness
		12.3. Bounded parabolic points
		12.4. Geometrically finite groups
			12.4.1. Characterizations of geometrical finiteness
			12.4.2. Consequences of geometrical finiteness
			12.4.3. Examples of geometrically finite groups
	Chapter 13. Counterexamples
		13.1. Embedding \\R-trees into real hyperbolic spaces
		13.2. Strongly discrete groups with infinite Poincaré exponent
		13.3. Moderately discrete groups which are not strongly discrete
		13.4. Poincaré irregular groups
		13.5. Miscellaneous counterexamples
	Chapter 14. \\R-trees and their isometry groups
		14.1. Construction of \\R-trees by the cone method
		14.2. Graphs with contractible cycles
		14.3. The nearest-neighbor projection onto a convex set
		14.4. Constructing \\R-trees by the stapling method
		14.5. Examples of \\R-trees constructed using the stapling method
Part  4 .  Patterson–Sullivan theory
	Chapter 15. Conformal and quasiconformal measures
		15.1. The definition
		15.2. Conformal measures
		15.3. Ergodic decomposition
		15.4. Quasiconformal measures
			15.4.1. Pointmass quasiconformal measures
			15.4.2. Non-pointmass quasiconformal measures
	Chapter 16. Patterson–Sullivan theorem for groups of divergence type
		16.1. Samuel–Smirnov compactifications
		16.2. Extending the geometric functions to \\what????
		16.3. Quasiconformal measures on \\what????
		16.4. The main argument
		16.5. End of the argument
		16.6. Necessity of the generalized divergence type assumption
		16.7. Orbital counting functions of nonelementary groups
	Chapter 17. Quasiconformal measures of geometrically finite groups
		17.1. Sufficient conditions for divergence type
		17.2. The global measure formula
		17.3. Proof of the global measure formula
		17.4. Groups for which ???? is doubling
		17.5. Exact dimensionality of ????
			17.5.1. Diophantine approximation on Λ
			17.5.2. Examples and non-examples of exact dimensional measures
	Appendix A. Open problems
	Appendix B. Index of defined terms
	Bibliography
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