Parametrisation of decorated Margulis spacetimes using strip deformations

Abstract

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.

Figures (12)

• Figure 1: An ideal pentagon
• Figure 2: A fundamental domain for an ideal one-holed square
• Figure 3: A fundamental domain for a (1,2)-spiked annulus
• Figure 4: A fundamental domain for a 3-spiked Möbius strip
• Figure 5: A metric on $S^{1,2,3}_{0,3}$

Theorems & Definitions (67)

• theorem 1
• theorem 2
• definition 1
• definition 2
• definition 3
• theorem 3
• definition 4
• definition 5
• definition 6
• theorem 4