Bounded Cohomology, a Rough Outline

Author(s):  
Riccardo Benedetti ◽  
Carlo Petronio
2018 ◽  
Vol 21 (2) ◽  
pp. 381-403 ◽  
Author(s):  
Jean-François Lafont ◽  
Shi Wang
Keyword(s):  

2018 ◽  
Vol 30 (5) ◽  
pp. 1157-1162 ◽  
Author(s):  
Michelle Bucher ◽  
Nicolas Monod

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.


2020 ◽  
Vol 66 (1) ◽  
pp. 151-172
Author(s):  
Clara Löh ◽  
Roman Sauer

Author(s):  
M. MORASCHINI ◽  
A. SAVINI

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.


2004 ◽  
Vol 2004 (40) ◽  
pp. 2103-2121 ◽  
Author(s):  
Igor V. Erovenko

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.


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

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