Margulis spacetimes via the arc complex

• 成果类型: Article
• 署名作者: Danciger, Jeffrey; Gueritaud, Francois; Kassel, Fanny
• 署名单位: University of Texas System; University of Texas Austin; Centre National de la Recherche Scientifique (CNRS); Universite de Lille
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-015-0610-z
• 发表日期: 2016
• 页码: 133-193
• 关键词: affine transformations; discontinuous groups; SURFACES; BOUNDARY; geometry; SPACES; HYPERBOLICITY; 3-manifolds; geodesics

摘要

We study strip deformations of convex cocompact hyperbolic surfaces, defined by inserting hyperbolic strips along a collection of disjoint geodesic arcs properly embedded in the surface. We prove that any deformation of the surface that uniformly lengthens all closed geodesics can be realized as a strip deformation, in an essentially unique way. The infinitesimal version of this result gives a parameterization, by the arc complex, of the moduli space of Margulis spacetimes with fixed convex cocompact linear holonomy. As an application, we provide a new proof of the tameness of such Margulis spacetimes M by establishing the Crooked Plane Conjecture, which states that M admits a fundamental domain bounded by piecewise linear surfaces called crooked planes. The noninfinitesimal version gives an analogous theory for noncompact complete anti-de Sitter 3-manifolds.