Proper affine actions and geodesic flows of hyperbolic surfaces

• 成果类型: Article
• 署名作者: Goldman, William M.; Labourie, Francois; Margulis, Gregory
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2009.170.1051
• 发表日期: 2009
• 页码: 1051-1083
• 关键词: strong morita equivalence; margulis space-times; discontinuous groups; linear part; transformations; isospectrality; CURVATURE; MANIFOLDS; ALGEBRAS; holonomy

摘要

Let Gamma(0) subset of O(2, 1) be a Schottky group, and let Sigma = H-2/Gamma(0) be the corresponding hyperbolic surface. Let C(Sigma) denote the space of unit length geodesic currents on Sigma. The cohomology group H-1(Gamma(0), V) parametrizes equivalence classes of affine deformations Gamma(u) of Gamma(0) acting on an irreducible representation V of O(2, 1). We define a continuous biaffine map psi : C(Sigma) x H-1 (Gamma(0), V) -> R which is linear on the vector space H-1 (Gamma(0), V). An affine deformation Gamma(u) acts properly if and only if psi(mu, [u]) not equal 0 for all mu is an element of 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), V) over the Fricke-Teichmuller space of Sigma.