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We extend Ghys theory about semiconjugacy to the world of measurable cocycles. More precisely, given a measurable cocycle with values into $text{Homeo}^+(mathbb{S}^1)$, we can construct a $text{L}^infty$-parametrized Euler class in bounded cohomology. We show that such a class vanishes if and only if the cocycle can be lifted to $text{Homeo}^+_{mathbb{Z}}(mathbb{R})$ and it admits an equivariant family of points. We define the notion of semicohomologous cocycles and we show that two measurable cocycles are semicohomologous if and only if they induce the same parametrized Euler class. Since for minimal cocycles, semicohomology boils down to cohomology, the parametrized Euler class is constant for minimal cohomologous cocycles. We conclude by studying the vanishing of the real parametrized Euler class and we obtain some results of elementarity.
Let $Gamma$ be a finitely generated group and let $(X,mu_X)$ be an ergodic standard Borel probability $Gamma$-space. Suppose that $G$ is the connected component of the identity of the isometry group of a Hermitian symmetric space. Given a Zariski den
In nature, one observes that a K-theory of an object is defined in two steps. First a structured category is associated to the object. Second, a K-theory machine is applied to the latter category to produce an infinite loop space. We develop a genera
This note proves that, as K-theory elements, the symbol classes of the de Rham operator and the signature operator on a closed manifold of even dimension are congruent mod 2. An equivariant generalization is given pertaining to the equivariant Euler characteristic and the multi-signature.
We study the large scale geometry of mapping class groups MCG(S), using hyperbolicity properties of curve complexes. We show that any self quasi-isometry of MCG(S) (outside a few sporadic cases) is a bounded distance away from a left-multiplication,
We present three new inequalities tying the signature, the simplicial volume and the Euler characteristic of surface bundles over surfaces. Two of them are true for any surface bundle, while the third holds on a specific family of surface bundles, na