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Bounded Cohomology and Simplicial Volume [Paperback / softback]

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Edited by , Edited by , Edited by (Universidad Autónoma de Madrid), Edited by (Universitŕ di Bologna)
  • Format: Paperback / softback, 170 pages, height x width x depth: 228x151x10 mm, weight: 260 g, Worked examples or Exercises
  • Series: London Mathematical Society Lecture Note Series
  • Pub. Date: 17-Nov-2022
  • Publisher: Cambridge University Press
  • ISBN-10: 100918329X
  • ISBN-13: 9781009183291
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Bounded Cohomology and Simplicial VolumeLarger Image
  • Format: Paperback / softback, 170 pages, height x width x depth: 228x151x10 mm, weight: 260 g, Worked examples or Exercises
  • Series: London Mathematical Society Lecture Note Series
  • Pub. Date: 17-Nov-2022
  • Publisher: Cambridge University Press
  • ISBN-10: 100918329X
  • ISBN-13: 9781009183291
Other books in subject:
Permanent link: https://www.kriso.lt/db/9781009183291.html
Keywords:
Since their introduction by Gromov in the 1980s, the study of bounded cohomology and simplicial volume has developed into an active field connected to geometry and group theory. This monograph, arising from a learning seminar for young researchers working in the area, provides a collection of different perspectives on the subject, both classical and recent. The book's introduction presents the main definitions of the theories of bounded cohomology and simplicial volume, outlines their history, and explains their principal motivations and applications. Individual chapters then present different aspects of the theory, with a focus on examples. Detailed references to foundational papers and the latest research are given for readers wishing to dig deeper. The prerequisites are only basic knowledge of classical algebraic topology and of group theory, and the presentations are gentle and informal in order to be accessible to beginning graduate students wanting to enter this lively and topical field.

This monograph presents an overview of bounded cohomology, simplicial volume, and related topics, covering the basics of the subject and recent research directions. Aimed at graduate students entering the field, it includes precise definitions and many concrete examples while keeping an informal style and minimising prerequisites.

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An overview of bounded cohomology and simplicial volume covering the basics of the subject and recent research directions.
List of Contributors
x
Preface xi
Acknowledgments xiii
Introduction xiv
Part I SIMPLICIAL VOLUME
1 Gromov's Mapping Theorem via Multicomplexes
3(17)
Marco Moraschini
1.1 Strategy of the Proof of Theorem 1.1
4(2)
1.2 Multicomplexes
6(2)
1.3 The Singular Multicomplex
8(2)
1.4 Complete Multicomplexes
10(3)
1.5 Minimal Multicomplexes
13(2)
1.6 Aspherical Multicomplexes
15(5)
2 The Proportionality Principle via Hyperbolic Geometry
20(8)
Filippo Sarti
2.1 Volume of Simplices in Hn
21(1)
2.2 Simplicial Volume
21(1)
2.3 Straightening of Simplices
22(2)
2.4 Efficient Cycles
24(2)
2.5 The Proof
26(2)
3 Positi vity of Simplicial Volume via Barycentric Techniques
28(15)
Shi Wang
3.1 Results and Examples
28(3)
3.2 Straightening and Local Straightening
31(2)
3.3 Barycentric Straightening
33(2)
3.4 Jacobian Estimates
35(8)
4 Gromov's Systolic Inequality via Smoothing
43(6)
Lizhi Chen
4.1 Gromov's Systolic Inequality
43(1)
4.2 Straight Invariant Fundamental Cocycles
44(1)
4.3 An Alternative Definition of Simplicial Volume
45(1)
4.4 The Smoothing Technique
46(2)
4.5 Applications of the Smoothing Technique
48(1)
5 Integral Foliated Simplicial Volume
49(8)
Caterina Campagnolo
5.1 A Question of Gromov
49(1)
5.2 Integral Simplicial Volume
50(1)
5.3 Stable Integral Simplicial Volume
51(2)
5.4 Integral Foliated Simplicial Volume
53(4)
6 L2-Betti Numbers
57(8)
Holger Kammeyer
6.1 The Definition of L2-Betti Numbers
57(2)
6.2 Some Properties of L2-Betti Numbers
59(2)
6.3 Relevance of L2-Betti Numbers
61(4)
Part II BOUNDED COHOMOLOGY
7 Stable Commutator Length
65(12)
Nicolaus Heuer
7.1 Three Ways to Stumble upon scl
65(5)
7.2 Vanishing, Gaps, and Lions
70(2)
7.3 Spectrum
72(2)
7.4 Relationship to Simplicial Volume
74(1)
7.5 Open Questions in scl
75(2)
8 Quasimorphisms on Negatively Curved Groups
77(8)
Biao Ma
8.1 Bounded Cohomology and Quasimorphisms
77(2)
8.2 Hyperbolic Groups and Mapping Class Groups
79(2)
8.3 WPD and Quasimorphisms
81(4)
9 Extension of Quasicocycles from Hyperbolically Embedded Subgroups
85(15)
Francesco Fournier-Facio
9.1 Alternating Quasicocycles
85(3)
9.2 Intuitive Idea
88(3)
9.3 Hyperbolically Embedded Subgroups
91(5)
9.4 The Trace Operator
96(2)
9.5 Implications
98(2)
10 Lie Groups and Symmetric Spaces
100(8)
Anton Hase
10.1 Symmetric Spaces
100(4)
10.2 Continuous (Bounded) Cohomology of Lie Groups
104(4)
11 Continuous Bounded Cohomology, Representations, and Multiplicative Constants
108(10)
Alessio Sa vini
11.1 Continuous Bounded Cohomology
108(1)
11.2 Measurable and Essentially Bounded Functions
109(4)
11.3 Multiplicative Constants
113(1)
11.4 Examples and Applications
114(4)
12 The Proportionality Principle via Bounded Cohomology Filippo Baroni
118(14)
12.1 Straightening in Non-positive Curvature
119(1)
12.2 Duality and the Volume Coclass
120(3)
12.3 Continuous Cohomology
123(2)
12.4 The Proportionality Principle
125(4)
12.5 Simplicial Volume of Hyperbolic Manifolds
129(3)
References 132(9)
Index 141
Caterina Campagnolo is a postdoctoral researcher now working at UAM Madrid. Francesco Fournier-Facio is PhD student at ETH Zürich. Nicolaus Heuer received his PhD from the University of Oxford. Marco Moraschini is a type A fixed-termed Researcher at University of Bologna. He was previously a Postdoctoral Researcher at University of Regensburg.