6 problems

Superrigidity conjecture. Maximal representations of a complex hyperbolic lattice into a Hermitian Lie group should be superrigid.

Let $\Gamma<\operatorname{SU}(1,n)$ be a lattice, with $n\geq 2$, and let $\rho:\Gamma\to\operatorname{PO}_{\mathbf R}(2,\infty)$ be a representation. Nonexistence conjecture. Ther…

Let $n\geq 2$, let $\Gamma<\operatorname{SU}(1,n)$ be a lattice, and let $\rho:\Gamma\to\operatorname{PO}_{\mathbf C}(p,\infty)$ be a maximal representation. The associated invaria…

Rigidity conjecture. If $\rho\colon\Gamma\to G$ is maximal, then $G=\operatorname{SU}(p,q)$ with $p\geq nq$, and $\rho$ is a trivial deformation of $\rho_{\mathrm{std}}$, namely

Twisted diagonal deformation conjecture. If $G$ is not locally isomorphic to $\mathrm{Sp}(2n,\mathbf{R})$, then every maximal representation

Topological invariants conjecture. If $G$ is not locally isomorphic to $\mathrm{Sp}(2n,\mathbf{R})$, then the topological invariants of Theorem distinguish connected components of…