Compact quotients, Anosov representations and sphere bundles

The so-called Problem of Compact Quotients asks for which homogeneous spaces G/H there exists a discrete subgroup Gamma of G such that Gamma\G/H is a compact manifold. We will discuss two methods for addressing this problem: a geometric one making links with Anosov representations, and a topological one based on weak triviality properties of certain sphere bundles. This is joint work with Nicolas Tholozan and Yosuke Morita.