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Proper affine actions and geodesic flows of hyperbolic surfaces

William Goldman·Francois Labourie·Gregory Margulis
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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.

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William Goldman, Francois Labourie, Gregory Margulis. 2004-06-12. Proper affine actions and geodesic flows of hyperbolic surfaces. https://arxiv.org/abs/math/0406247

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