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ONE-DIMENSIONAL SUBGROUPS AND CONNECTED COMPONENTS IN NON-ABELIAN p-ADIC DEFINABLE GROUPS

Part of: Model theory

Published online by Cambridge University Press:  29 April 2024

WILLIAM JOHNSON Open the ORCID record for WILLIAM JOHNSON [Opens in a new window]  and
NINGYUAN YAO
WILLIAM JOHNSON*
Affiliation:
SCHOOL OF PHILOSOPHY FUDAN UNIVERSITY 220 HANDAN ROAD GUANGHUA WEST BUILDING ROOM 2503 SHANGHAI 200433 CHINA E-mail: yaony@fudan.edu.cn
NINGYUAN YAO
Affiliation:
SCHOOL OF PHILOSOPHY FUDAN UNIVERSITY 220 HANDAN ROAD GUANGHUA WEST BUILDING ROOM 2503 SHANGHAI 200433 CHINA E-mail: yaony@fudan.edu.cn
*
E-mail: willjohnson@fudan.edu.cn
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Abstract

We generalize two of our previous results on abelian definable groups in p-adically closed fields [12, 13] to the non-abelian case. First, we show that if G is a definable group that is not definably compact, then G has a one-dimensional definable subgroup which is not definably compact. This is a p-adic analogue of the Peterzil–Steinhorn theorem for o-minimal theories [16]. Second, we show that if G is a group definable over the standard model $\mathbb {Q}_p$, then $G^0 = G^{00}$. As an application, definably amenable groups over $\mathbb {Q}_p$ are open subgroups of algebraic groups, up to finite factors. We also prove that $G^0 = G^{00}$ when G is a definable subgroup of a linear algebraic group, over any model.

Keywords

definable groupsp-adically closed fieldsNIP groups

MSC classification

Primary: 03C60: Model-theoretic algebra
Type
Article
Information
The Journal of Symbolic Logic , First View , pp. 1 - 19
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of The Association for Symbolic Logic

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