Abstract

Discrete subgroups of Lie groups play a fundamental role in several areas of mathematics. In the case of $\mathrm{SL}(2,\mathbb{R})$, they are well understood and classified by the geometry of the corresponding hyperbolic surfaces. In the case of $\mathrm{SL}(n,\mathbb{R})$ with $n>2$, they remain more mysterious, beyond the important class of lattices (i.e. discrete subgroups of finite covolume for the Haar measure). These past twenty years, several interesting classes of discrete subgroups have emerged, which are "thinner" than lattices, more flexible, and with remarkable geometric and dynamical properties. We will give an overview of such developments and present some of these new classes. We will also discuss when discrete subgroups can act properly discontinuously on homogeneous spaces, with an emphasis on the so-called Problem of Compact Quotients, which asks for which homogeneous spaces $G/H$ there exists a discrete subgroup $\Gamma$ of $G$ such that $\Gamma\backslash G/H$ is a compact manifold.