Proper affine actions and geodesic flows of hyperbolic surfaces

William M. Goldman¹ ; François Labourie² ; Gregory Margulis³

¹ Department of Mathematics<br/>University of Maryland<br/>College Park, MD 20742<br/>United States
² Topologie et Dynamique<br/>Université Paris-Sud<br/>F-91405 Orsay (Cedex)<br/>France
³ Department of Mathematics<br/>Yale University<br/>10 Hillhouse Ave.<br/>P.O. Box 208283<br/>New Haven, CT 06520<br/>United States

Annals of mathematics, Tome 170 (2009) no. 3, pp. 1051-1083

Résumé

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 $\Sigma$.

Détail

DOI : 10.4007/annals.2009.170.1051