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Register a new userA Matsumoto-Mostow result for Zimmers cocycles of hyperbolic lattices
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As for the theory of maximal representations, we introduce the volume of a Zimmers cocycle $Gamma times X rightarrow mbox{PO}^circ(n, 1)$, where $Gamma$ is a torsion-free (non-)uniform lattice in $mbox{PO}^circ(n, 1)$, with $n geq 3$, and $X$ is a suitable standard Borel probability $Gamma$-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 $Gamma backslash mathbb{H}^n$. This invariant can be interpreted as a suitable multiplicative constant between bounded cohomology classes. This allows us to characterize maximal cocycles for being cohomologous to the cocycle induced by the standard lattice embedding via a measurable map $X rightarrow mbox{PO}(n, 1)$ with essentially constant sign. As a by-product of our rigidity result for the volume of cocycles, we give a new 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 closed surface groups. It extends the classic Euler number of representations and it agrees with the generalized version of the Euler number of self-couplings up to a multiplicative constant. We show a Milnor-Wood type inequality whose upper bound is given by the modulus of the Euler characteristic. This gives an alternative proof of the same result for the generalized version of the Euler number of self-couplings. Finally, we characterize maximal cocycles as those which are cohomologous to the one induced by a hyperbolization.
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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.
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.
We prove that any action of a higher rank lattice on a Gromov-hyperbolic space is elementary. More precisely, it is either elliptic or parabolic. This is a large generalization of the fact that any action of a higher rank lattice on a tree has a fixed point. A consequence is that any quasi-action of a higher rank lattice on a tree is elliptic, i.e. it has Mannings property (QFA). Moreover, we obtain a new proof of the theorem of Farb-Kaimanovich-Masur that any morphism from a higher rank lattice to a mapping class group has finite image, without relying on the Margulis normal subgroup theorem nor on bounded cohomology. More generally, we prove that any morphism from a higher rank lattice to a hierarchically hyperbolic group has finite image. In the Appendix, Vincent Guirardel and Camille Horbez deduce rigidity results for morphisms from a higher rank lattice to various outer automorphism groups.
We classify representations of a class of Deligne-Mostow lattices into PGL(3, C). In particular, we show local rigidity for the representations (of Deligne-Mostow lattices with 3-fold symmetry and of type one) where the generators we chose are in the same conjugacy class as the generators of Deligne-Mostow lattices. We use formal computations in SAGE to obtain the results. The code files are available on GitHub ([FPUP21]).
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