Functorial Free Group from Anosov Representations on Bundles

Let $\rho :\mathrm{\Gamma }\to G$ be an Anosov representation, with $\mathrm{\Gamma }$ 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 ${\mathrm{\Omega }}_{\rho }\subset G\mathrm{/}H$, where $H$ is a parabolic or symmetric subgroup. In this paper, we extend the properly discontinuous $\mathrm{\Gamma }$-action via $\rho$ to the space of connections on the pullbacks of the tangent bundle over ${\mathrm{\Omega }}_{\rho }$. When ${\mathrm{\Omega }}_{\rho }$ is a complex curve, we show that the $\mathrm{\Gamma }$-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 $\rho (\mathrm{\Gamma })$ acts properly discontinuously on $F^{ab}\setminus \{\mathrm{i}\mathrm{d}\}$. 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 ${\mathrm{\Omega }}_{\rho }$ and construct a natural functor to the category of abelian groups.