Definable Equivariant Retractions in Non-Archimedean GeometryOpen Access

Hils, Martin; Hrushovski, Ehud; Simon, Pierre

Research article in digital collection | Preprint

Abstract

For G an algebraic group definable over a model of ACVF, or more generally a definable subgroup of an algebraic group, we study the stable completion Gˆ of G, as introduced by Loeser and the second author. For G connected and stably dominated, assuming G commutative or that the valued field is of equicharacteristic 0, we construct a pro-definable G-equivariant strong deformation retraction of Gˆ onto the generic type of G. For G=S a semiabelian variety, we construct a pro-definable S-equivariant strong deformation retraction of Sˆ onto a definable group which is internal to the value group. We show that, in case S is defined over a complete valued field K with value group a subgroup of R, this map descends to an S(K)-equivariant strong deformation retraction of the Berkovich analytification San of S onto a piecewise linear group, namely onto the skeleton of San. This yields a construction of such a retraction without resorting to an analytic (non-algebraic) uniformization of S. Furthermore, we prove a general result on abelian groups definable in an NIP theory: any such group G is a directed union of ∞-definable subgroups which all stabilize a generically stable Keisler measure on G.

Details about the publication

Name of the repositoryarxiv
Article number2101.02619
Statussubmitted / under review
Release year2021
Language in which the publication is writtenEnglish
DOI10.48550/arXiv.2101.02619
Link to the full texthttps://arxiv.org/abs/2101.02619
KeywordsModelltheorie; Nichtarchimedische Geometrie

Authors from the University of Münster

Hils, Martin
Professorship for Mathematical Logic (Prof. Hils)

Projects the publication originates from

Duration: 01/01/2020 - 31/12/2024
Funded by: DFG - Individual Grants Programme
Type of project: Individual project
Duration: 01/01/2019 - 31/12/2025 | 1st Funding period
Funded by: DFG - Cluster of Excellence
Type of project: Subproject in DFG-joint project hosted at University of Münster
Duration: 01/09/2016 - 30/06/2019 | 1st Funding period
Funded by: DFG - Collaborative Research Centre
Type of project: Subproject in DFG-joint project hosted at University of Münster

Vorträge zur Publikation

Equivariant Definable Deformation Retractions in Non-archimedean Geometry
Hils, Martin (29/07/2019)
"Model Theory Seminar", University of Waterloo, Waterloo, Kanada
Type of talk: scientific talk
Geometric Reductions in Algebraically Closed Valued Fields
Hils, Martin (08/07/2019)
"Oberseminar Mathematische Logik", Albert-Ludwigs-Universität Freiburg, Freiburg, Deutschland
Type of talk: scientific talk
Definable equivariant retractions onto skeleta in non-Archimedean geometry
Hils, Martin (28/03/2018)
"Model Theory and Applications" (final conference of the trimester "Model Theory, Combinatorics and Valued Fields"), Institut Henri Poincaré, Paris, Frankreich
Type of talk: scientific talk