Non-Archimedean Hilbert geometry and degenerations of real Hilbert geometries
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866915217946968064 |
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| author | Flamm, Xenia Parreau, Anne |
| author_facet | Flamm, Xenia Parreau, Anne |
| contents | We develop a theory of Hilbert geometry over general ordered valued fields, associating with an open convex subset of the projective space a quotient Hilbert metric space. Under natural non-degeneracy assumptions, we prove that the ultralimit of a sequence of rescaled real Hilbert geometries is isometric to the Hilbert metric space of an open convex projective subset over a Robinson field. This result allows us to prove that ideal points of the space of convex real projective structures on a closed manifold arise from actions on non-Archimedean Hilbert geometries without global fixed point. We explicitly describe the Hilbert metric space of a non-Archimedean bounded polytope $P$ defined over a subfield of the valuation ring as the geometric realization of the flag complex of $P$ modeled on a Weyl chamber. As an application, we obtain a complete description of Gromov-Hausdorff limits of a real polytope with rescaled Hilbert metric. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_22671 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Non-Archimedean Hilbert geometry and degenerations of real Hilbert geometries Flamm, Xenia Parreau, Anne Metric Geometry Differential Geometry Geometric Topology 53C60, 53C15, 54A20, 12J15, 12J25 We develop a theory of Hilbert geometry over general ordered valued fields, associating with an open convex subset of the projective space a quotient Hilbert metric space. Under natural non-degeneracy assumptions, we prove that the ultralimit of a sequence of rescaled real Hilbert geometries is isometric to the Hilbert metric space of an open convex projective subset over a Robinson field. This result allows us to prove that ideal points of the space of convex real projective structures on a closed manifold arise from actions on non-Archimedean Hilbert geometries without global fixed point. We explicitly describe the Hilbert metric space of a non-Archimedean bounded polytope $P$ defined over a subfield of the valuation ring as the geometric realization of the flag complex of $P$ modeled on a Weyl chamber. As an application, we obtain a complete description of Gromov-Hausdorff limits of a real polytope with rescaled Hilbert metric. |
| title | Non-Archimedean Hilbert geometry and degenerations of real Hilbert geometries |
| topic | Metric Geometry Differential Geometry Geometric Topology 53C60, 53C15, 54A20, 12J15, 12J25 |
| url | https://arxiv.org/abs/2503.22671 |