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تسجيل مستخدم جديدMultiplicative constants and maximal measurable cocycles in bounded cohomology
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Multiplicative constants are a fundamental tool in the study of maximal representations. In this paper we show how to extend such notion, and the associated framework, to measurable cocycles theory. As an application of this approach, we define and study the Cartan invariant for measurable $textup{PU}(m,1)$-cocycles of complex hyperbolic lattices.
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Let $Gamma$ be a torsion-free lattice of $text{PU}(p,1)$ with $p geq 2$ and let $(X,mu_X)$ be an ergodic standard Borel probability $Gamma$-space. We prove that any maximal Zariski dense measurable cocycle $sigma: Gamma times X longrightarrow text{SU
}(m,n)$ is cohomologous to a cocycle associated to a representation of $text{PU}(p,1)$ into $text{SU}(m,n)$, with $1 < m leq n$. The proof follows the line of Zimmer Superrigidity Theorem and requires the existence of a boundary map, that we prove in a much more general setting. As a consequence of our result, it cannot exist a maximal measurable cocycle with the above properties when $n eq m$.
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Following the work of Burger, Iozzi and Wienhard for representations, in this paper we introduce the notion of maximal measurable cocycles of a surface group. More precisely, let $mathbf{G}$ be a semisimple algebraic $mathbb{R}$-group such that $G=ma
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