Discrete subgroups of semisimple Lie groups, beyond lattices

Discrete subgroups of SL(2,R) are well understood, and classified by the geometry of the corresponding hyperbolic surfaces. Discrete subgroups of higher-rank semisimple Lie groups, such as SL(n,R) for n>2, remain more mysterious. While lattices in this setting are rigid, there also exist more flexible, "thinner" discrete subgroups, which may have large and interesting deformation spaces, giving rise in particular to so-called higher Teichm\"uller theory. We survey recent progress in constructing and understanding such discrete subgroups from a geometric and dynamical viewpoint.