Boundary maps and maximal representations on infinite-dimensional Hermitian symmetric spaces

We define a Toledo number for actions of surface groups and complex hyperbolic lattices on infinite-dimensional Hermitian symmetric spaces, which allows us to define maximal representations. When the target is not of tube type, we show that there cannot be Zariski-dense maximal representations, and whenever the existence of a boundary map can be guaranteed, the representation preserves a finite-dimensional totally geodesic subspace on which the action is maximal. In the opposite direction, we construct examples of geometrically dense maximal representation in the infinite-dimensional Hermitian symmetric space of tube type and finite rank. Our approach is based on the study of boundary maps, which we are able to construct in low ranks or under some suitable Zariski density assumption, circumventing the lack of local compactness in the infinite-dimensional setting.

Multiplicative constants and maximal measurable cocycles in bounded cohomology

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 $\mathrm{PU}(m,1)$ -cocycles of complex hyperbolic lattices.

Superrigidity of maximal measurable cocycles of complex hyperbolic lattices

Let $$\Gamma $$ Γ be a torsion-free lattice of $$PU (p,1)$$ P U ( p , 1 ) with $$p \ge 2$$ p ≥ 2 and let $$(X,\mu _X)$$ ( X , μ X ) be an ergodic standard Borel probability $$\Gamma $$ Γ -space. We prove that any maximal Zariski dense measurable cocycle $$\sigma : \Gamma \times X \longrightarrow SU (m,n)$$ σ : Γ × X ⟶ S U ( m , n ) is cohomologous to a cocycle associated to a representation of $$PU (p,1)$$ P U ( p , 1 ) into $$SU (m,n)$$ S U ( m , n ) , with $$1 \le m \le n$$ 1 ≤ m ≤ 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, there cannot exist maximal measurable cocycles with the above properties when $$1< m < n$$ 1 < m < n .

Hyperbolic topology and bounded locally homeomorphic quasiregular mappings in 3-space

We use our new type of bounded locally homeomorphic quasiregular mappings in the unit 3-ball to address long standing problems for such mappings, including the Vuorinen injectivity problem. The construction of such mappings comes from our construction of non-trivial compact 4-dimensional cobordisms $M$ with symmetric boundary components and whose interiors have complete 4-dimensional real hyperbolic structures. Such bounded locally homeomorphic quasiregular mappings are defined in the unit 3-ball $B^3\subset \mathbb{R}^3$ as mappings equivariant with the standard conformal action of uniform hyperbolic lattices $\Gamma\subset \operatorname{Isom} H^3$ in the unit 3-ball and with its discrete representation $G=\rho(\Gamma)\subset \operatorname{Isom} H^4 $. Here, $G$ is the fundamental group of our non-trivial hyperbolic 4-cobordism $M=(H^4\cup\Omega(G))/G,$ and the kernel of the homomorphism $\rho\!:\! Gamma\rightarrow G$ is a free group $F_3$ on three generators.