Georgia Institute of Technology College of Sciences

A sequence of remarkable results in recent decades have shown that for a surface group H there are many Lie groups G and connected components C of Hom(H,G) consisting of discrete and faithful representations. These are known as higher Teichmüller spaces. With two exceptions, all known constructions of higher Teichmüller spaces work only for surface groups. This is an expository talk on the remarkable paper Convexes Divisibles III (Benoist ‘05), in which the first construction of higher Teichmüller spaces that works for some non-surface-groups was discovered. The paper implies the fundamental group H’ of any closed hyperbolic n-manifold has a higher Teichmüller space C’ in PGL(n+1,R). This is proved by showing any element of C’ preserves a convex domain in RP^n with a group-invariant tiling.