On Vinberg's notion of quasi-arithmeticity

In the late 60s, Vinberg introduced his notion of quasi-arithmeticity, a weakening of the notion of arithmeticity for lattices in semisimple Lie groups. It follows from the superrigidity theorems of Margulis and Gromov–Schoen, and from work of Esnault–Groechenig, that the existence of quasi-arithmetic lattices that are not in fact arithmetic is essentially a real hyperbolic phenomenon. I will explain how quasi-arithmeticity arises naturally in the study of arithmetic (real) hyperbolic manifolds themselves, and is indeed a useful tool for distinguishing commensurability classes of non-(quasi-)arithmetic hyperbolic manifolds. Time permitting, I will also discuss some open questions