Minimal submanifolds, low dimensional actions and higher expansion phenomena

In a recent work with Ben Lowe we studied the behavior of low codimensional minimal submanifolds in locally symmetric spaces. This led us to discover a curious dichotomy, where (under some reasonable conditions on the ambient space) any low codimensional subset of a locally symmetric space has to be “very complicated” or otherwise it can be collapsed to lower dimension. This proved to be surprisingly useful observation, which we can exploit to prove branched cover stability theorems for octonionic hyperbolic manifolds, show new fixed point theorems for low dimensional action of lattices in semisimple Lie groups and build new examples of Riemannian higher expanders in the sense of Gromov. I will explain how minimal submanifolds and geometric measure theory tools enter the picture and advertise several new ways they could be used to study locally symmetric spaces.