Fundamental domains in photon space

A representation of a surface group into a Lie group of Hermitian type G is called maximal if it maximizes the Toledo invariant. In a 2017 joint work with N. Treib, we proved that in the case of surfaces with non-empty boundary, all such representations could be obtained by a Ping-pong construction in the Shilov boundary of G. For the specific case of Sp(2n,R), maximal representations admit open cocompact domains of discontinuity in the projective space, and we used the Ping-pong construction to obtain explicit fundamental domains for this action bounded by quadric hypersurfaces. In joint work with Pier-Olivier Rodrigue, we construct similar domains for the action of maximal representations in SO(2,n) on the grassmannian of isotropic 2-planes (also called photon space, as it parametrizes photons in the Einstein Universe). We also prove that the same hypersurfaces can be used to bound fundamental domains for representations obtained by the inclusion of a classical Schottky group in SO(1,n) into SO(2,n) and their small deformations.