Domains of discontinuity for (quasi-)Hitchin representations
Amphithéâtre Léon Motchane (IHES)
Amphithéâtre Léon Motchane
Among representations of surface groups into Lie groups, the Anosov representations are the ones with the nicest dynamical properties.
Guichard-Wienhard and Kapovich-Leeb-Porti have shown that their actions on generalized flag manifolds often admit co-compact domains of discontinuity, whose quotients are closed manifolds carrying interesting geometric structures.
Dumas and Sanders studied the topology and the geometry of the quotient in the case of quasi-Hitchin representations (Anosov representations which are deformations of Hitchin representations). In a conjecture they ask whether these manifolds are homeomorphic to fiber bundles over the surface.
In joint work with Qiongling Li, we can prove that the conjecture is true for (quasi-)Hitchin representations in SL(n,R) and SL(n,C), acting on projective spaces and partial flag manifolds parametrizing points and hyperplanes.