Domain of Discontinuity of Anosov Subgroups Acting on Bundles

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 \. 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.