ON GROUPS and FIELDS DEFINABLE in 1-H-MINIMAL FIELDS

Juan Pablo Acosta López; Assaf Hasson

doi:10.1017/S1474748024000239

ON GROUPS and FIELDS DEFINABLE in 1-H-MINIMAL FIELDS

Juan Pablo Acosta López, Assaf Hasson

Department of Mathematics

Ben-Gurion University of the Negev

Research output: Contribution to journal › Article › peer-review

Abstract

We show that an infinite group G definable in a 1-h-minimal field admits a strictly K-differentiable structure with respect to which G is a (weak) Lie group, and we show that definable local subgroups sharing the same Lie algebra have the same germ at the identity. We conclude that infinite fields definable in K are definably isomorphic to finite extensions of K and that 1-dimensional groups definable in K are finite-by-Abelian-by-finite. Along the way, we develop the basic theory of definable weak K-manifolds and definable morphisms between them.

Original language	American English
Journal	Journal of the Institute of Mathematics of Jussieu
DOIs	https://doi.org/10.1017/S1474748024000239
State	Accepted/In press - 1 Jan 2024

Keywords

groups
lie groups
model theory
valuation

All Science Journal Classification (ASJC) codes

General Mathematics

Access to Document

10.1017/S1474748024000239

Cite this

@article{8a9be4fbdb2343b7855d4493592ca514,

title = "ON GROUPS and FIELDS DEFINABLE in 1-H-MINIMAL FIELDS",

abstract = "We show that an infinite group G definable in a 1-h-minimal field admits a strictly K-differentiable structure with respect to which G is a (weak) Lie group, and we show that definable local subgroups sharing the same Lie algebra have the same germ at the identity. We conclude that infinite fields definable in K are definably isomorphic to finite extensions of K and that 1-dimensional groups definable in K are finite-by-Abelian-by-finite. Along the way, we develop the basic theory of definable weak K-manifolds and definable morphisms between them.",

keywords = "groups, lie groups, model theory, valuation",

author = "{Acosta L{\'o}pez}, {Juan Pablo} and Assaf Hasson",

note = "Publisher Copyright: {\textcopyright} The Author(s), 2024.",

year = "2024",

month = jan,

day = "1",

doi = "10.1017/S1474748024000239",

language = "American English",

journal = "Journal of the Institute of Mathematics of Jussieu",

issn = "1474-7480",

publisher = "Cambridge University Press",

}

TY - JOUR

T1 - ON GROUPS and FIELDS DEFINABLE in 1-H-MINIMAL FIELDS

AU - Acosta López, Juan Pablo

AU - Hasson, Assaf

PY - 2024/1/1

Y1 - 2024/1/1

N2 - We show that an infinite group G definable in a 1-h-minimal field admits a strictly K-differentiable structure with respect to which G is a (weak) Lie group, and we show that definable local subgroups sharing the same Lie algebra have the same germ at the identity. We conclude that infinite fields definable in K are definably isomorphic to finite extensions of K and that 1-dimensional groups definable in K are finite-by-Abelian-by-finite. Along the way, we develop the basic theory of definable weak K-manifolds and definable morphisms between them.

AB - We show that an infinite group G definable in a 1-h-minimal field admits a strictly K-differentiable structure with respect to which G is a (weak) Lie group, and we show that definable local subgroups sharing the same Lie algebra have the same germ at the identity. We conclude that infinite fields definable in K are definably isomorphic to finite extensions of K and that 1-dimensional groups definable in K are finite-by-Abelian-by-finite. Along the way, we develop the basic theory of definable weak K-manifolds and definable morphisms between them.

KW - groups

KW - lie groups

KW - model theory

KW - valuation

UR - http://www.scopus.com/inward/record.url?scp=85195063243&partnerID=8YFLogxK

U2 - 10.1017/S1474748024000239

DO - 10.1017/S1474748024000239

M3 - Article

SN - 1474-7480

JO - Journal of the Institute of Mathematics of Jussieu

JF - Journal of the Institute of Mathematics of Jussieu

ER -

ON GROUPS and FIELDS DEFINABLE in 1-H-MINIMAL FIELDS

Abstract

Keywords

All Science Journal Classification (ASJC) codes

Access to Document

Other files and links

Cite this