Functorial Free Group from Anosov Representations on Bundles

TL;DR

This work extends the properly discontinuous action of Anosov representations from the finite dimensional domain Omega_rho to associated infinite dimensional geometric spaces built from pullbacks of the tangent bundle, including spaces of forms and connections and, when Omega_rho is a complex curve, Higgs bundles. It defines a free abelian group F^{ab} generated by holomorphic line bundles arising from pullback tangent data and proves that Gamma acts properly discontinuously on F^{ab} minus the identity, with F^{ab} well defined up to isomorphism on the Zariski-dense Anosov character variety. The paper further introduces a categorical structure on Anosov representations and constructs a covariant functor to Ab sending each representation to its F^{ab}, demonstrating naturality of the construction with respect to morphisms. These results provide new dynamical invariants and a functorial framework for higher Teichmuller type representations.

Abstract

Let $ρ: Γ\to G$ be an Anosov representation, with $Γ$ 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 $Ω_ρ\subset G/H$, 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 pullbacks of the tangent bundle over $Ω_ρ$. When $Ω_ρ$ is a complex curve, we show that the $Γ$-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 $ρ(Γ)$ 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 $Ω_ρ$ and construct a natural functor to the category of abelian groups.

Theorems & Definitions (18)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Theorem 1.5
• Theorem 2.1
• Definition 2.2
• Remark 2.3
• Lemma 3.1
• Proposition 3.2