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E-grāmata: Flexibility of Group Actions on the Circle

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  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 2231
  • Izdošanas datums: 02-Jan-2019
  • Izdevniecība: Springer Nature Switzerland AG
  • Valoda: eng
  • ISBN-13: 9783030028558
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Flexibility of Group Actions on the CirclePalielināt
  • Formāts: PDF+DRM
  • Sērija : Lecture Notes in Mathematics 2231
  • Izdošanas datums: 02-Jan-2019
  • Izdevniecība: Springer Nature Switzerland AG
  • Valoda: eng
  • ISBN-13: 9783030028558
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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.

Recenzijas

The book contains a lot of information and is written in a concise and well-organized way, describing its contents in a long introductory chapter and starting each chapter with a small abstract, with many references to the relevant literature and comments on related concepts and on related work (maybe sometimes the addition of more intuitive versions and motivations of some of the basic, sometimes quite technical definitions and concepts would have been helpful for a less experienced reader). (Bruno Zimmermann, zbMATH 1407.57001, 2019)

1 Introduction
1(18)
1.1 Some Basic Notions
1(2)
1.2 Combination Theorems and Indiscrete Subgroups of PSL-2 (R)
3(2)
1.3 Uncountable Families of Exotic Group Actions on the Circle
5(3)
1.4 An Axiomatic Approach to Combination Theorems
8(1)
1.5 Flexibility and Rigidity
9(1)
1.6 Mapping Class Groups
10(1)
1.7 Notes and References
10(6)
1.7.1 Circle Actions and Quasimorphisms
10(1)
1.7.2 Generalizations to Other Semi-simple Algebraic Groups
11(1)
1.7.3 Towards a Teichmuller Theory for Indiscrete Representations
11(1)
1.7.4 Dense Limit Subgroups of Algebraic Groups
12(1)
1.7.5 Relationship to the Work of Calegari and Calegari-Walker
13(1)
1.7.6 Dense Sets of Faithful Projective Surface Group Actions
13(1)
1.7.7 Projective Actions Versus Analytic Actions
14(1)
1.7.8 Non-Fuchsian Exotic Actions
15(1)
1.7.9 Groups Without Exotic Actions
15(1)
1.7.10 Mapping Class Groups
16(1)
1.8 Outline of the Monograph
16(3)
2 Preliminaries
19(16)
2.1 Actions on the Circle
19(9)
2.1.1 Rotation Number and Euler Class
20(1)
2.1.2 Semi-conjugacy
21(3)
2.1.3 Limit Set of a Circle Action
24(1)
2.1.4 Blow-Up and Minimalization
25(2)
2.1.5 Minimal Quasimorphisms
27(1)
2.2 Hyperbolic Geometry
28(5)
2.2.1 Maximal Abelian Subgroups
28(2)
2.2.2 Finite Type Hyperbolic 2-Orbifolds
30(1)
2.2.3 Commutative-Transitive Groups
31(1)
2.2.4 Minimalization of Fuchsian Groups
32(1)
2.3 Indiscrete Subgroups of PSL2(R)
33(2)
3 Topological Baumslag Lemmas
35(10)
3.1 Topological Setting
35(4)
3.2 Projective and Discrete Settings
39(6)
4 Splittable Fuchsian Groups
45(26)
4.1 Very General Points, Abundance and Stable Injectivity
45(3)
4.2 Pulling-Apart Lemma
48(7)
4.3 Almost Faithful Paths
55(9)
4.4 Simultaneous Control of Rotation Numbers
64(7)
5 Combination Theorem for Flexible Groups
71(10)
5.1 Statement of the Result
71(2)
5.2 Proof
73(2)
5.3 Exotic Circle Actions
75(2)
5.4 Quasimorphisms
77(1)
5.5 Limit Groups
78(3)
6 Axiomatics
81(12)
6.1 Tracial Structures
81(4)
6.2 UV-Structures
85(3)
6.3 Combination Theorem for Smooth Actions
88(5)
7 Mapping Class Groups
93(4)
7.1 The Universal Circle and Nielsen's Action
93(2)
7.2 Exotic Mapping Class Group Actions
95(2)
8 Zero Rotation Spectrum and Teichmuller Theory
97(18)
8.1 Rigidity of Projective Actions
97(3)
8.2 Lie Subgroups of the Circle Homeomorphism Group
100(5)
8.3 Free and Surface Subgroups of π1(M)
105(7)
8.3.1 Quasi-Fuchsian Surface Subgroups
106(2)
8.3.2 Geometrically Infinite Surface Groups
108(2)
8.3.3 Free Subgroups of π1(M)
110(2)
8.4 Nonlinear Smooth Actions of Free Groups
112(3)
A Equivalent Notions of Semi-conjugacy 115(12)
References 127(6)
Index 133
Sang-hyun Kim received his PhD from Yale University in 2007, under the supervision of Andrew J. Casson. He previously worked at the University of Texas at Austin, Tufts University and KAIST. He is a member of the Young Korean Academy of Science and Technology (Y-KAST), and a recipient of the Sang-San Prize for Young Mathematicians. He is currently an Associate Professor at Seoul National University.





Thomas Koberda received his PhD in 2012 from Harvard University, under the supervision of Curtis T. McMullen. He was an NSF Postdoctoral Fellow and a Gibbs Assistant Professor at Yale University. He is currently on the faculty of the University of Virginia. In 2017, he was named an Alfred P. Sloan Foundation Research Fellow, and was awarded the Kamil Duszenko Prize.





Mahan Mj received his PhD from the University of California at Berkeley in 1997, under the supervision of Andrew J. Casson. He is currently Professor of Mathematics at Tata Institute of Fundamental Research. He was awarded the Infosys Prize in Mathematical Sciences in 2015 and was an invited speaker in the Geometry Section at the International Congress of Mathematicians, 2018.
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