7 problems

Let a lattice in $SU(1,n)$ act on the boundary of complex hyperbolic space. Boundary-action rigidity question. Is the action of any lattice in $SU(1,n)$ on the boundary of complex…

Let $\rho$ be an arithmetic isometric action of a lattice in $SU(1,n)$. Arithmetic-action deformation conjecture. Any sufficiently small perturbation of $\rho$ is of the form obtai…

Let $G=G_1\times\cdots\times G_k$ be a semisimple Lie group with all factors non-compact, and let $\Lambda<H$ be a cocompact lattice, with $G<H$. A left translation action $\rho$ o…

Let $G=G_1\times\cdots\times G_k$ be a semisimple Lie group whose factors are all non-compact, let $\Gamma<G$ be an irreducible lattice, and let $\rho$ be an affine action of…

Let $G$ be a semi-simple Lie group with no compact factors and no simple factors isomorphic to $SO(1,n)$ or $SU(1,n)$, and let $\Gamma<G$ be a lattice. A generalized quasi-affine a…

Let $G$ be a 1-connected nilpotent Lie group, $H \subset G$ a closed subgroup, and $\Gamma \subset G$ a non-trivial discrete subgroup. Let $\mathcal{R}(\Gamma,G,H)$ denote the spac…

Local measure-rigidity conjecture. The action $\alpha'$ has an ergodic invariant measure $\mu'$ satisfying the same assumptions; $\mu'$ can be chosen so that $\mu'\to\mu$ in the we…