Proper affine actions and geodesic flows of hyperbolic surfaces

Abstract

Let $\Gamma_0\subset\mathsf{O}(2,1)$ be a Schottky group, and let $\Sigma=\mathsf{H}^2/\Gamma_0$ be the corresponding hyperbolic surface. Let $\mathcal{C}(\Sigma)$ denote the space of unit length geodesic currents on $\Sigma$. The cohomology group $H^1(\Gamma_0,\mathsf{V})$ parametrizes equivalence classes of affine deformations $\Gamma_{\mathsf u}$ of $\Gamma_0$ acting on an irreducible representation $\mathsf{V}$ of $\mathsf{O}(2,1)$. We define a continuous biaffine map $\Psi: \mathcal{C}(\Sigma) \times H^1(\Gamma_0,\mathsf{V}) \rightarrow\mathbb{R} $ which is linear on the vector space $H^1(\Gamma_0,\mathsf{V})$. An affine deformation $\Gamma_{\mathsf u}$ acts properly if and only if $\Psi(\mu,[\mathsf{u}])\neq 0$ for all $\mu\in\mathcal{C}(\Sigma)$. Consequently the set of proper affine actions whose linear part is a Schottky group identifies with a bundle of open convex cones in $H^1(\Gamma_0,\mathsf{V})$ over the Fricke-Teichmüller space of $\Sigma$.