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Register a new userAlgebraic hull of maximal measurable cocycles of surface groups into Hermitian Lie groups
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Authors
Alessio Savini
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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 $G=mathbf{G}(mathbb{R})^circ$ 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.
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Let $G$ a semisimple Lie group of non-compact type and let $mathcal{X}_G$ be the Riemannian symmetric space associated to it. Suppose $mathcal{X}_G$ has dimension $n$ and it has no factor isometric to either $mathbb{H}^2$ or $text{SL}(3,mathbb{R})/text{SO}(3)$. Given a closed $n$-dimensional Riemannian manifold $N$, let $Gamma=pi_1(N)$ be its fundamental group and $Y$ its universal cover. Consider a representation $rho:Gamma rightarrow G$ with a measurable $rho$-equivariant map $psi:Y rightarrow mathcal{X}_G$. Connell-Farb described a way to construct a map $F:Yrightarrow mathcal{X}_G$ which is smooth, $rho$-equivariant and with uniformly bounded Jacobian. In this paper we extend the construction of Connell-Farb to the context of measurable cocycles. More precisely, if $(Omega,mu_Omega)$ is a standard Borel probability $Gamma$-space, let $sigma:Gamma times Omega rightarrow G$ be a measurable cocycle. We construct a measurable map $F: Y times Omega rightarrow mathcal{X}_G$ which is $sigma$-equivariant, whose slices are smooth and they have uniformly bounded Jacobian. For such equivariant maps we define also the notion of volume and we prove a sort of mapping degree theorem in this particular context.
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 $n eq m$.
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.
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.
We generalize the theory of the second invariant cohomology group $H^2_{rm inv}(G)$ for finite groups $G$, developed in [Da2,Da3,GK], to the case of affine algebraic groups $G$, using the methods of [EG1,EG2,G]. In particular, we show that for connected affine algebraic groups $G$ over an algebraically closed field of characteristic $0$, the map $Theta$ from [GK] is bijective (unlike for some finite groups, as shown in [GK]). This allows us to compute $H^2_{rm inv}(G)$ in this case, and in particular show that this group is commutative (while for finite groups it can be noncommutative, as shown in [GK]).
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