Proper affine actions and geodesic flows of hyperbolic surfaces

By William M. Goldman, F. Labourie, Gregory Margulis. 2004

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Let To C 0(2,1) be a Schottky group, and let ? = H2/ To be the corresponding hyperbolic surface. Let ^(S) denote the space of unit length geodesic currents on S. The cohomology group if1 (To, V) parametrizes equivalence classes of affine deformations Tu of To acting on an irreducible representation V of 0(2,1). We define a continuous biaffine map *I> : ^(S) x Hl(To, V) ?> U which is linear on the vector space H1(Fo, V). An affine deformation Tu acts properly if and only if *(/x, [u]) 7^ 0 for all /jl e ^(S). Consequently the set of proper affine actions whose linear part is a Schottky group identifies with a bundle of open convex cones in Hl (To, V) over the Fricke-Teichmiiller space of E.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/87eb4c9dc274e2dfd1050917acac4b5fa12327a6