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Schottky spaces and universal Mumford curves over Z
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Poineau, Jérôme and Turchetti, Daniele (2022) Schottky spaces and universal Mumford curves over Z. Selecta Mathematica, New Series, 28 (4). 79. doi:10.1007/s00029-022-00793-z ISSN 1022-1824.
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WRAP-Schottky-spaces-and-universal-Mumford-curves-over-Z-Turchettie-2022.pdf - Accepted Version - Requires a PDF viewer. Download (918Kb) | Preview |
Official URL: http://dx.doi.org/10.1007/s00029-022-00793-z
Abstract
For every integer g≥1 we define a universal Mumford curve of genus g in the framework of Berkovich spaces over Z. This is achieved in two steps: first, we build an analytic space Sg that parametrizes marked Schottky groups over all valued fields. We show that Sg is an open, connected analytic space over Z. Then, we prove that the Schottky uniformization of a given curve behaves well with respect to the topology of Sg, both locally and globally. As a result, we can define the universal Mumford curve Cg as a relative curve over Sg such that every Schottky uniformized curve can be described as a fiber of a point in Sg. We prove that the curve Cg is itself uniformized by a universal Schottky group acting on the relative projective line P1Sg. Finally, we study the action of the group Out(Fg) of outer automorphisms of the free group with g generators on Sg, describing the quotient Out(Fg)∖Sg in the archimedean and non-archimedean cases. We apply this result to compare the non-archimedean Schottky space with constructions arising from geometric group theory and the theory of moduli spaces of tropical curves.
| Item Type: | Journal Article | ||||||
|---|---|---|---|---|---|---|---|
| Subjects: | Q Science > QA Mathematics | ||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||
| Library of Congress Subject Headings (LCSH): | Riemann surfaces, Topology | ||||||
| Journal or Publication Title: | Selecta Mathematica, New Series | ||||||
| Publisher: | Springer ; Birkhäuser Verlag | ||||||
| ISSN: | 1022-1824 | ||||||
| Official Date: | 2 September 2022 | ||||||
| Dates: |
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| Volume: | 28 | ||||||
| Number: | 4 | ||||||
| Article Number: | 79 | ||||||
| DOI: | 10.1007/s00029-022-00793-z | ||||||
| Status: | Peer Reviewed | ||||||
| Publication Status: | Published | ||||||
| Reuse Statement (publisher, data, author rights): | “This version of the article has been accepted for publication, after peer review (when applicable) but is not the Version of Record and does not reflect post-acceptance improvements, or any corrections. The Version of Record is available online at: http://dx.doi.org/10.1007/s00029-022-00793-z. Use of this Accepted Version is subject to the publisher’s Accepted Manuscript terms of use https://www.springernature.com/gp/open-research/policies/acceptedmanuscript-terms”." | ||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||
| Date of first compliant deposit: | 6 September 2022 | ||||||
| Date of first compliant Open Access: | 2 September 2023 | ||||||
| Open Access Version: |
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