2023/06/16 by Pallavi Panda, Panda, Pallavi
Mathematics · #Differential Geometry (math.DG) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometric Topology (math.GT) #Geometric and Algebraic Topology
paper · pdf · doi:10.48550/arxiv.2306.09985
openalex publication_date 2023/06/16 · openalex created_date 2023/06/20 · openalex updated_date 2026/07/28
Margulis spacetimes are complete affine 3-manifolds that were introduced to show that the cocompactness condition of Auslander's conjecture is necessary. There are Lorentzian manifolds that are obtained as a quotient of the three dimensional Minkowski space by a non-abelian free group acting properly discontinuously by affine isometries. Goldman-Labourie-Margulis showed that such a group is determined by a complete hyperbolic metric on a possibly non-orientable finite-type hyperbolic surface together with an infinitesimal deformation of this metric that uniformly lengthens all non-trivial closed curves on the surface. Furthermore, the set of all such infinitesimal deformations forms an open convex cone. Danciger Guéritaud-Kassel parametrised the moduli space of Margulis spacetimes, with a fixed convex cocompact linear part, using the pruned arc complex. The parametrisation is done by gluing infinitesimal hyperbolic strips along a family of embedded, pairwise disjoint arcs of the hyperbolic surface that decompose it into topological disks. We generalise this result to complete finite-area hyperbolic surfaces with spikes decorated with horoballs. These are closely related to Margulis spacetimes decorated with finitely many pairwise disjoint affine light-like lines, called photons.