Functorial Free Group from Anosov Representations on Bundles

Let $\rho: \Gamma \to G$ be an Anosov representation, with $\Gamma$ a word hyperbolic group and $G$ a semisimple Lie group. Previous works (Guichard--Wienhard, Kapovich--Leeb--Porti, and Carvajales--Stecker) constructed an open domain of discontinuity $\Omega_\rho \subset G/H$, where $H$ is a parabolic or symmetric subgroup. In this paper, we extend the properly discontinuous $\Gamma$-action via $\rho$ to the space of connections on the pullbacks of the tangent bundle over $\Omega_\rho$. When $\Omega_\rho$ is a complex curve, we show that the $\Gamma$-action is properly discontinuous on the union of Higgs bundle structures associated with the $(1,0)$ part of the complexified pullback bundles. We further construct a free abelian group $F^{ab}$ generated by these holomorphic line bundles and induce a topoogical structure on it, so that $\rho(\Gamma)$ acts properly discontinuously on $F^{ab} \setminus \{\mathrm{id}\}$. This free abelian group is 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 compatible with $\Omega_\rho$ and construct a natural functor to the category of abelian groups.