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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.
List of contents
Introduction Caterina Campagnolo, Francesco Fournier-Facio, Nicolaus Heuer, Marco Moraschini; Part I. Simplicial Volume: 1. Gromov's mapping theorem via multicomplexes Marco Moraschini; 2. The proportionality principle via hyperbolic geometry Filippo Sarti; 3. Positivity of simplicial volume via barycentric techniques Shi Wang; 4. Gromov's systolic inequality via smoothing Lizhi Chen; Integral foliated simplicial volume Caterina Campagnolo; 6. l2-Betti numbers Holger Kammeyer; Part II. Bounded Cohomology: 7. Stable commutator length Nicolaus Heuer; 8. Quasimorphisms on negatively curved groups Biao Ma; 9. Extension of quasicocycles from hyperbolically embedded subgroups Francesco Fournier-Facio; 10. Lie groups and Symmetric spaces Anton Hase; 11. Continuous bounded cohomology, representations and multiplicative constants Alessio Savini; 12. The proportionality principle via bounded Filippo Baroni; References; Index.
About the author
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.
Summary
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.
