Structure and Regularity of Group Actions on One-Manifolds (Springer Monographs in Mathematics)

Preface
Acknowledgements
Contents
Chapter 1 Introduction
	1.0.1 Some conventions and notation
	1.1 Groups of Manifold Diffeomorphisms
	1.2 The Mather–Thurston Theorem, the Epstein–Ling Theorem, and Lattice-Like Rigidity
	1.3 The Takens–Filipkiewicz Theorem and Rubin’s Theorem
	1.4 Critical Regularity: History and Overview
		1.4.1 Definitions and remarks
		1.4.2 Cyclic groups, topological versus algebraic critical regularity
		1.4.3 Foliations
		1.4.4 Nilpotent groups
		1.4.5 Right-angled Artin groups and mapping class groups
		1.4.6 Property (T) and intermediate growth
		1.4.7 Lipschitz lower bounds
		1.4.8 Groups of prescribed critical regularity
	1.5 What This Book is About
	1.6 What This Book is Not About
	1.7 What We Will Assume of the Reader
Chapter 2 Denjoy’s Theorem and Exceptional Diffeomorphisms of the Circle
	2.1 The Minimal Set and the Rotation Number
		2.1.1 Minimal sets and exceptional diffeomorphisms
		2.1.2 The rotation number
		2.1.3 Rotation numbers, invariant measures, and amenability
		2.1.4 Invariant measures and free subgroups
	2.2 Denjoy’s Theorem
		2.2.1 Rational approximations of real numbers
		2.2.2 The Denjoy–Koksma inequality and unique ergodicity
		2.2.3 Completing the proof of Denjoy’s Theorem
	2.3 Exceptional Diffeomorphisms and Integrable Moduli
		2.3.1 Stationary measures and Lipschitz homeomorphisms
		2.3.2 Smooth Denjoy counterexamples and the spectrum of their moduli of continuity
Chapter 3 Full Diffeomorphism Groups Determine the Diffeomorphism Class of a Manifold
	3.1 Diffeomorphism Groups of General Manifolds and Critical Regularity
	3.2 Generalities From Differential Topology
		3.2.1 Open balls in manifolds
		3.2.2 Fragmentation of diffeomorphisms
	3.3 Simplicity of Commutator Subgroups
		3.3.1 Higman’s Theorem
		3.3.2 The Epstein–Ling Theorem
	3.4 The Bochner–Montgomery Theorem on Continuous Group Actions
		3.4.1 Analytic prerequisites
		3.4.2 Standing assumptions
		3.4.3 On spatial derivatives
		3.4.4 On time derivatives
	3.5 Takens’ Theorem
		3.5.1 Promoting a bijection to a homeomorphism
		3.5.2 Takens’ theorem via Bochner–Montgomery
		3.5.3 Addendum: Takens’ original proof
	3.6 Rubin’s Theorem
		3.6.1 First-order expressibility of rigid stabilizers
		3.6.2 From groups to Boolean algebras
		3.6.3 From Boolean algebras to topologies
		3.6.4 Applications to manifold homeomorphism groups
		3.6.5 Locally moving groups obey no law
	3.7 The Original Proof of Whittaker–Filipkiewicz
		3.7.1 Transitivity, dilativity and weak fragmentation
		3.7.2 The pre-stabilizer subgroup
		3.7.3 Reconstructing a homeomorphism from pre-stabilizer groups
Chapter 4 The C^1 and C^2 Theory of Diffeomorphism Groups
	4.1 Kopell’s Lemma
	4.2 (Residually) Nilpotent Groups Acting By C^1 Diffeomorphisms
		4.2.1 The Farb–Franks Theorem and universal nilpotent groups
		4.2.2 Topological conjugacy of virtually nilpotent group actions
	4.3 The Two-Jumps Lemma and the abt-Lemma
		4.3.1 The Two-jumps lemma
		4.3.2 The abt-Lemma
	4.4 Crossed Homeomorphisms and Commutation
	4.5 Groups of C^1+bv Diffeomorphisms
	4.6 Classifying C^1-Actions of Solvable Baumslag–Solitar Groups
		4.6.1 Topological smoothing of the standard affine action
		4.6.2 C^1-actions on intervals
		4.6.3 C^1-actions on circles
Chapter 5 Chain Groups
	5.1 Chains and Covering Distances
	5.2 Generalities on Chain Groups
		5.2.1 Two-chain groups and Thompson’s group F
		5.2.2 Smooth realizations of Thompson’s group
		5.2.3 Subgroups of chain groups
	5.3 Simplicity
	5.4 The Chain Group Trick and the Rank Trick
		5.4.1 Maximal rank and the chain group trick
		5.4.2 The rank trick
Chapter 6 The Slow Progress Lemma
	6.1 Statement of the Result
	6.2 Natural Densities
	6.3 Probabilistic Dynamical Behavior
	6.4 Proof of the Slow Progress Lemma
Chapter 7 Algebraic Obstructions for General Regularities
	7.1 Statement of the Results
	7.2 Strategy for the Proof
	7.3 A Single Diffeomorphism of Optimal Expansion
	7.4 Construction of Optimally Expanding Diffeomorphism Groups
		7.4.1 Specifying the images of generators
		7.4.2 Verifying the optimal expansion of φ
	7.5 Promotion to Circles
	7.6 Continua of GroupsWith Prescribed Critical Hölder Regularity
Chapter 8 Applications
	8.1 Foliation Theory
		8.1.1 Foliations and suspensions of group actions
		8.1.2 Non-smoothability of codimension one foliations
	8.2 Right-Angled Artin Groups
		8.2.1 Abelian groups, free groups, and smooth actions
		8.2.2 P_4, the cograph hierarchy, and C^1+bv actions
	8.3 Mapping Class Groups
		8.3.1 Continuous actions
		8.3.2 Smoothing to C^2 and beyond and connections to right-angled Artin groups
		8.3.3 Smooth actions of related groups
		8.3.4 C^1+τ actions
		8.3.5 C^1 actions and Ivanov’s Conjecture
Appendix A Concave Moduli of Continuity
	A.1 Smooth Optimal Concave Moduli
	A.2 The Group Structure on Diff^k,α (M)
	A.3 The Muller–Tsuboi Trick for Non-Integer Regularities
Appendix B Orderability and Hölder’s Theorem
	B.1 Orderability of Groups and Homeomorphism Groups
		B.1.1 Linear orderability and the interval
		B.1.2 Circular orderability and the circle
	B.2 Hölder’s Theorem
Appendix C The Thurston Stability Theorem
References
Index