Proper affine actions and geodesic flows of hyperbolic surfaces
Differential Geometry
2011-07-12 v3 Dynamical Systems
Abstract
We give necessary and sufficient conditions for an affine deformation of a Schottky subgroup of O(2,1) to act properly on affine space. There exists a real-valued biaffine map between the cohomology of the Schottky group and the space of geodesic currents on the corresponding hyperbolic surface S. For a fixed cohomology class, this map is uniformly positive or uniformly negative on the space of geodesic currents if and only if the corresponding affine deformation is proper. As a corollary, the deformation space of proper affine deformations is an open convex cone.
Cite
@article{arxiv.math/0406247,
title = {Proper affine actions and geodesic flows of hyperbolic surfaces},
author = {William Goldman and Francois Labourie and Gregory Margulis},
journal= {arXiv preprint arXiv:math/0406247},
year = {2011}
}
Comments
Several revisions, according to referee's suggestions. Simplifications in the proof in the last two sections. Accepted for publication in Annals of Mathematics