Flexibility of Group Actions on the Circle (Lecture Notes in Mathematics Book 2231)

In this partly expository work, a framework is developed for building exotic circle actions of certain classical groups. The authors give general combination theorems for indiscrete isometry groups of hyperbolic space which apply to Fuchsian and limit groups. An abundance of integer-valued subadditive defect-one quasimorphisms on these groups follow as a corollary. The main classes of groups considered are limit and Fuchsian groups. Limit groups are shown to admit large collections of faithful actions on the circle with disjoint rotation spectra. For Fuchsian groups, further flexibility results are proved and the existence of non-geometric actions of free and surface groups is established. An account is given of the extant notions of semi-conjugacy, showing they are equivalent. This book is suitable for experts interested in flexibility of representations, and for non-experts wanting an introduction to group representations into circle homeomorphism groups. Preface......Page 6 Acknowledgements......Page 7 Contents......Page 8 1.1 Some Basic Notions......Page 10 1.2 Combination Theorems and Indiscrete Subgroups of PSL2(R)......Page 12 1.3 Uncountable Families of Exotic Group Actions on the Circle......Page 14 1.4 An Axiomatic Approach to Combination Theorems......Page 17 1.5 Flexibility and Rigidity......Page 18 1.7.1 Circle Actions and Quasimorphisms......Page 19 1.7.3 Towards a Teichmüller Theory for Indiscrete Representations......Page 20 1.7.4 Dense Limit Subgroups of Algebraic Groups......Page 21 1.7.6 Dense Sets of Faithful Projective Surface Group Actions......Page 22 1.7.7 Projective Actions Versus Analytic Actions......Page 23 1.7.9 Groups Without Exotic Actions......Page 24 1.8 Outline of the Monograph......Page 25 2.1 Actions on the Circle......Page 27 2.1.1 Rotation Number and Euler Class......Page 28 2.1.2 Semi-conjugacy......Page 29 2.1.3 Limit Set of a Circle Action......Page 32 2.1.4 Blow-Up and Minimalization......Page 33 2.1.5 Minimal Quasimorphisms......Page 35 2.2.1 Maximal Abelian Subgroups......Page 36 2.2.2 Finite Type Hyperbolic 2-Orbifolds......Page 38 2.2.3 Commutative–Transitive Groups......Page 39 2.2.4 Minimalization of Fuchsian Groups......Page 40 2.3 Indiscrete Subgroups of PSL2(R)......Page 41 3.1 Topological Setting......Page 43 3.2 Projective and Discrete Settings......Page 47 4.1 Very General Points, Abundance and Stable Injectivity......Page 53 4.2 Pulling-Apart Lemma......Page 56 4.3 Almost Faithful Paths......Page 63 4.4 Simultaneous Control of Rotation Numbers......Page 72 5.1 Statement of the Result......Page 78 5.2 Proof......Page 80 5.3 Exotic Circle Actions......Page 82 5.4 Quasimorphisms......Page 84 5.5 Limit Groups......Page 85 6.1 Tracial Structures......Page 87 6.2 UV-Structures......Page 91 6.3 Combination Theorem for Smooth Actions......Page 94 7.1 The Universal Circle and Nielsen's Action......Page 99 7.2 Exotic Mapping Class Group Actions......Page 101 8.1 Rigidity of Projective Actions......Page 103 8.2 Lie Subgroups of the Circle Homeomorphism Group......Page 106 8.3 Free and Surface Subgroups of π1(M)......Page 111 8.3.1 Quasi-Fuchsian Surface Subgroups......Page 112 8.3.2 Geometrically Infinite Surface Groups......Page 114 8.3.3 Free Subgroups of π1(M)......Page 116 8.4 Nonlinear Smooth Actions of Free Groups......Page 118 A Equivalent Notions of Semi-conjugacy......Page 121 References......Page 132 Index......Page 137 "In this partly expository work, the authors develop a framework for building exotic circle actions of certain classical groups. The authors give general combination theorems for indiscrete isometry groups of hyperbolic space which apply to Fuchsian and limit groups. An abundance of integer-valued subadditive defect-one quasimorphisms on these groups follow as a corollary. The main classes of groups considered in this book are limit and Fuchsian groups. Limit groups are shown to admit large collections of faithful actions on the circle with disjoint rotation spectra. For Fuchsian groups, the authors prove further flexibility results and show the existence of non-geometric actions of free and surface groups. The authors give an accound of the extant notions of semi-conjugacy, showing they are equivalent. This book is suitable for an expert interested in flexibility of representations, and for a non-expert wanting an introduction to group representations into circle homeomorphism groups"--Page 4 of cover