Proper affine actions and geodesic flows of hyperbolic surfaces
نویسنده
چکیده
Let Γ0 ⊂ O(2, 1) be a Schottky group, and let Σ = H2/Γ0 be the corresponding hyperbolic surface. Let C(Σ) denote the space of geodesic currents on Σ. The cohomology group H1(Γ0, V) parametrizes equivalence classes of affine deformations Γu of Γ0 acting on an irreducible representation V of O(2, 1). We define a continuous biaffine map C(Σ)×H1(Γ0, V) Ψ −→ R which is linear on the vector space H1(Γ0, V). An affine deformation Γu acts properly if and only if Ψ(μ, [u]) 6= 0 for all μ ∈ C(Σ). Consequently the set of proper affine actions whose linear part is a Schottky group identifies with a bundle of open convex cones in H1(Γ0, V) over the Teichmüller space of Σ.
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