Let ρ: Γ G be an Anosov representation, with Γ a word hyperbolic group and G a semisimple Lie group. Previous works (by Guichard--Wienhard, Kapovich--Leeb--Porti, and recently Carvajales--Stecker) constructed an open domain of discontinuity Ω_ρ G/H, where H is a parabolic or symmetric subgroup. In this paper, we extend the properly discontinuous Γ-action (via ρ) to the space of connections on the pullback of the tangent bundle over Ω_ρ. When Ω_ρ is a complex surface, we show that the Γ-action is properly discontinuous on the union of the Higgs bundle structures of the (1, 0) part of the complexification of these pullback bundles. We further construct a topological free group F generated by these holomorphic line bundles and show that ρ (Γ) acts properly discontinuously on F \id\. This free group is shown to be well-defined up to isomorphism over the character variety of Zariski dense Anosov representations. Finally, we endow the space of Anosov representations with a categorical structure and construct a natural functor to the category of free groups.
Gongopadhyay et al. (Wed,) studied this question.