Bounded Cohomology, a Rough Outline

1992 ◽  
pp. 273-320
Author(s):  
Riccardo Benedetti ◽  
Carlo Petronio
Keyword(s):  
Bounded Cohomology ◽  
Rough Outline

scholarly journals Barycentric straightening and bounded cohomology

2018 ◽  
Vol 21 (2) ◽  
pp. 381-403 ◽  
Author(s):  
Jean-François Lafont ◽  
Shi Wang
Keyword(s):  
Bounded Cohomology

scholarly journals The cup product of Brooks quasimorphisms

2018 ◽  
Vol 30 (5) ◽  
pp. 1157-1162 ◽  
Author(s):  
Michelle Bucher ◽  
Nicolas Monod
Keyword(s):  
Free Group ◽  
Automorphism Groups ◽  
Bounded Cohomology ◽  
Universal Class ◽  
Cup Product

AbstractWe prove the vanishing of the cup product of the bounded cohomology classes associated to any two Brooks quasimorphisms on the free group. This is a consequence of the vanishing of the square of a universal class for tree automorphism groups.

scholarly journals Bounded cohomology of amenable covers via classifying spaces

2020 ◽  
Vol 66 (1) ◽  
pp. 151-172
Author(s):  
Clara Löh ◽  
Roman Sauer
Keyword(s):  
Classifying Spaces ◽  
Bounded Cohomology

scholarly journals A MATSUMOTO–MOSTOW RESULT FOR ZIMMER’S COCYCLES OF HYPERBOLIC LATTICES

2020 ◽  
Author(s):  
M. MORASCHINI ◽  
A. SAVINI
Keyword(s):  
Euler Number ◽  
Type Inequality ◽  
Surface Group ◽  
Closed Surface ◽  
Complete Characterization ◽  
Alternative Proof ◽  
Bounded Cohomology ◽  
Rigidity Result ◽  
Multiplicative Constant ◽  
Wood Type

AbstractFollowing the philosophy behind the theory of maximal representations, we introduce the volume of a Zimmer’s cocycle Γ × X → PO° (n, 1), where Γ is a torsion-free (non-)uniform lattice in PO° (n, 1), with n > 3, and X is a suitable standard Borel probability Γ-space. Our numerical invariant extends the volume of representations for (non-)uniform lattices to measurable cocycles and in the uniform setting it agrees with the generalized version of the Euler number of self-couplings. We prove that our volume of cocycles satisfies a Milnor–Wood type inequality in terms of the volume of the manifold Γ\ℍn. Additionally this invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to define a family of measurable cocycles with vanishing volume. The same interpretation enables us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map X → PO° (n, 1) with essentially constant sign.As a by-product of our rigidity result for the volume of cocycles, we give a different proof of the mapping degree theorem. This allows us to provide a complete characterization of maps homotopic to local isometries between closed hyperbolic manifolds in terms of maximal cocycles.In dimension n = 2, we introduce the notion of Euler number of measurable cocycles associated to a closed surface group and we show that it extends the classic Euler number of representations. Our Euler number agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. Imitating the techniques developed in the case of the volume, we show a Milnor–Wood type inequality whose upper bound is given by the modulus of the Euler characteristic of the associated closed surface. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, using the interpretation of the Euler number as a multiplicative constant between bounded cohomology classes, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.

scholarly journals On bounded cohomology of amalgamated products of groups

2004 ◽  
Vol 2004 (40) ◽  
pp. 2103-2121 ◽  
Author(s):  
Igor V. Erovenko
Keyword(s):  
Cohomology Group ◽  
Initial Segment ◽  
Singular Part ◽  
Bounded Cohomology ◽  
Products Of Groups ◽  
Amalgamated Products ◽  
Cohomology Sequence

We investigate the structure of the singular part of the second bounded cohomology group of amalgamated products of groups by constructing an analog of the initial segment of the Mayer-Vietoris exact cohomology sequence for the spaces of pseudocharacters.

Foundations of the theory of bounded cohomology

1987 ◽  
Vol 37 (3) ◽  
pp. 1090-1115 ◽  
Author(s):  
N. V. Ivanov
Keyword(s):  
Bounded Cohomology
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