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Register a new userMultiplicative 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 $n eq m$.
Given $Gamma < text{PU}(n,1)$ a torsion-free lattice and $(X,mu_X)$ a standard Borel $Gamma$-space, we introduce the notion of Toledo invariant of a measurable cocycle $sigma:Gamma times X rightarrow text{PU}(p,infty)$. Since that invariant has bounded absolute value, it makes sense to speak about maximality. We prove that any maximal measurable cocycle is finitely reducible, that is it admits a cohomologous cocycle with image contained in a copy of $text{PU}(p,np)$ inside $text{PU}(p,infty)$, which is a finite algebraic subgroup. Even if we do not provide a real rigidity result in this setting, our statement can be seen as the natural adaption of the results for representations due to Duchesne, Lecureux and Pozzetti. We conclude the paper by completing the analysis of maximal cocycles of complex hyperbolic lattices started in cite{sarti:savini} with a characterization of their algebraic hull.
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 $G=mathbf{G}(mathbb{R})^circ$ is of Hermitian type. If $Gamma leq L$ is a torsion-free lattice of a finite connected covering of $text{PU}(1,1)$, given a standard Borel probability $Gamma$-space $(Omega,mu_Omega)$, we introduce the notion of Toledo invariant for a measurable cocycle $sigma:Gamma times Omega rightarrow G$. The Toledo remains unchanged along $G$-cohomology classes and its absolute value is bounded by the rank of $G$. This allows to define maximal measurable cocycles. We show that the algebraic hull $mathbf{H}$ of a maximal cocycle $sigma$ is reductive and the centralizer of $H=mathbf{H}(mathbb{R})^circ$ is compact. If additionally $sigma$ admits a boundary map, then $H$ is of tube type and $sigma$ is cohomologous to a cocycle stabilizing a unique maximal tube-type subdomain. This result is analogous to the one obtained for representations. In the particular case $G=text{PU}(n,1)$ maximality is sufficient to prove that $sigma$ is cohomologous to a cocycle preserving a complex geodesic. We conclude with some remarks about boundary maps of maximal Zariski dense cocycles.
Let $text{G}(n)$ be equal either to $text{PO}(n,1),text{PU}(n,1)$ or $text{PSp}(n,1)$ and let $Gamma leq text{G}(n)$ be a uniform lattice. Denote by $mathbb{H}^n_K$ the hyperbolic space associated to $text{G}(n)$, where $K$ is a division algebra over the reals of dimension $d=dim_{mathbb{R}} K$. Assume $d(n-1) geq 2$. In this paper we generalize natural maps to measurable cocycles. Given a standard Borel probability $Gamma$-space $(X,mu_X)$, we assume that a measurable cocycle $sigma:Gamma times X rightarrow text{G}(m)$ admits an essentially unique boundary map $phi:partial_infty mathbb{H}^n_K times X rightarrow partial_infty mathbb{H}^m_K$ whose slices $phi_x:mathbb{H}^n_K rightarrow mathbb{H}^m_K$ are atomless for almost every $x in X$. Then, there exists a $sigma$-equivariant measurable map $F: mathbb{H}^n_K times X rightarrow mathbb{H}^m_K$ whose slices $F_x:mathbb{H}^n_K rightarrow mathbb{H}^m_K$ are differentiable for almost every $x in X$ and such that $text{Jac}_a F_x leq 1$ for every $a in mathbb{H}^n_K$ and almost every $x in X$. The previous properties allow us to define the natural volume $text{NV}(sigma)$ of the cocycle $sigma$. This number satisfies the inequality $text{NV}(sigma) leq text{Vol}(Gamma backslash mathbb{H}^n_K)$. Additionally, the equality holds if and only if $sigma$ is cohomologous to the cocycle induced by the standard lattice embedding $i:Gamma rightarrow text{G}(n) leq text{G}(m)$, modulo possibly a compact subgroup of $text{G}(m)$ when $m>n$. Given a continuous map $f:M rightarrow N$ between compact hyperbolic manifolds, we also obtain an adaptation of the mapping degree theorem to this context.
This paper is devoted to the construction of norm-preserving maps between bounded cohomology groups. For a graph of groups with amenable edge groups we construct an isometric embedding of the direct sum of the bounded cohomology of the vertex groups in the bounded cohomology of the fundamental group of the graph of groups. With a similar technique we prove that if (X,Y) is a pair of CW-complexes and the fundamental group of each connected component of Y is amenable, the isomorphism between the relative bounded cohomology of (X,Y) and the bounded cohomology of X in degree at least 2 is isometric. As an application we provide easy and self-contained proofs of Gromov Equivalence Theorem and of the additivity of the simplicial volume with respect to gluings along pi_1-injective boundary components with amenable fundamental group.
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