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| Natura: | Preprint |
| Pubblicazione: |
2023
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| Accesso online: | https://arxiv.org/abs/2306.09985 |
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| _version_ | 1866916118943236096 |
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| author | Panda, Pallavi |
| author_facet | Panda, Pallavi |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2306_09985 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Parametrisation of decorated Margulis spacetimes using strip deformations Panda, Pallavi Geometric Topology Differential Geometry 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. |
| title | Parametrisation of decorated Margulis spacetimes using strip deformations |
| topic | Geometric Topology Differential Geometry |
| url | https://arxiv.org/abs/2306.09985 |