Bounded Cohomology of Discrete Groups

Bounded Cohomology of Discrete Groups
Author: Roberto Frigerio
Publisher: American Mathematical Soc.
Total Pages: 213
Release: 2017-11-21
Genre: Mathematics
ISBN: 1470441462

The theory of bounded cohomology, introduced by Gromov in the late 1980s, has had powerful applications in geometric group theory and the geometry and topology of manifolds, and has been the topic of active research continuing to this day. This monograph provides a unified, self-contained introduction to the theory and its applications, making it accessible to a student who has completed a first course in algebraic topology and manifold theory. The book can be used as a source for research projects for master's students, as a thorough introduction to the field for graduate students, and as a valuable landmark text for researchers, providing both the details of the theory of bounded cohomology and links of the theory to other closely related areas. The first part of the book is devoted to settling the fundamental definitions of the theory, and to proving some of the (by now classical) results on low-dimensional bounded cohomology and on bounded cohomology of topological spaces. The second part describes applications of the theory to the study of the simplicial volume of manifolds, to the classification of circle actions, to the analysis of maximal representations of surface groups, and to the study of flat vector bundles with a particular emphasis on the possible use of bounded cohomology in relation with the Chern conjecture. Each chapter ends with a discussion of further reading that puts the presented results in a broader context.

Continuous Bounded Cohomology of Locally Compact Groups

Continuous Bounded Cohomology of Locally Compact Groups
Author: Nicolas Monod
Publisher: Springer
Total Pages: 219
Release: 2003-07-01
Genre: Mathematics
ISBN: 3540449620

Recent research has repeatedly led to connections between important rigidity questions and bounded cohomology. However, the latter has remained by and large intractable. This monograph introduces the functorial study of the continuous bounded cohomology for topological groups, with coefficients in Banach modules. The powerful techniques of this more general theory have successfully solved a number of the original problems in bounded cohomology. As applications, one obtains, in particular, rigidity results for actions on the circle, for representations on complex hyperbolic spaces and on Teichmüller spaces. A special effort has been made to provide detailed proofs or references in quite some generality.

Bounded Cohomology and Simplicial Volume

Bounded Cohomology and Simplicial Volume
Author: Caterina Campagnolo
Publisher: Cambridge University Press
Total Pages: 171
Release: 2022-11-30
Genre: Mathematics
ISBN: 100918329X

An overview of bounded cohomology and simplicial volume covering the basics of the subject and recent research directions.

Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups

Continuous Cohomology, Discrete Subgroups, and Representations of Reductive Groups
Author: Armand Borel
Publisher: American Mathematical Soc.
Total Pages: 282
Release: 2013-11-21
Genre: Mathematics
ISBN: 147041225X

It has been nearly twenty years since the first edition of this work. In the intervening years, there has been immense progress in the use of homological algebra to construct admissible representations and in the study of arithmetic groups. This second edition is a corrected and expanded version of the original, which was an important catalyst in the expansion of the field. Besides the fundamental material on cohomology and discrete subgroups present in the first edition, this edition also contains expositions of some of the most important developments of the last two decades.

On the Algebraic Foundations of Bounded Cohomology

On the Algebraic Foundations of Bounded Cohomology
Author: Theo Bühler
Publisher: American Mathematical Soc.
Total Pages: 126
Release: 2011
Genre: Mathematics
ISBN: 0821853112

It is a widespread opinion among experts that (continuous) bounded cohomology cannot be interpreted as a derived functor and that triangulated methods break down. The author proves that this is wrong. He uses the formalism of exact categories and their derived categories in order to construct a classical derived functor on the category of Banach $G$-modules with values in Waelbroeck's abelian category. This gives us an axiomatic characterization of this theory for free, and it is a simple matter to reconstruct the classical semi-normed cohomology spaces out of Waelbroeck's category. The author proves that the derived categories of right bounded and of left bounded complexes of Banach $G$-modules are equivalent to the derived category of two abelian categories (one for each boundedness condition), a consequence of the theory of abstract truncation and hearts of $t$-structures. Moreover, he proves that the derived categories of Banach $G$-modules can be constructed as the homotopy categories of model structures on the categories of chain complexes of Banach $G$-modules, thus proving that the theory fits into yet another standard framework of homological and homotopical algebra.

Geometry, Topology, and Dynamics in Negative Curvature

Geometry, Topology, and Dynamics in Negative Curvature
Author: C. S. Aravinda
Publisher: Cambridge University Press
Total Pages: 378
Release: 2016-01-21
Genre: Mathematics
ISBN: 110752900X

Ten high-quality survey articles provide an overview of important recent developments in the mathematics surrounding negative curvature.