One-dimensional subgroups and connected components in non-abelian p-adic definable groups

2023/08/03 by Johnson, Will, Yao, Ningyuan · 1 citation
#03C60 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.2308.01527

Abstract

We generalize two of our previous results on abelian definable groups in p-adically closed fields 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. Second, we show that if G is a group definable over the standard model ℚ_p, then G⁰ = G⁰⁰. As an application, definably amenable groups over ℚ_p are open subgroups of algebraic groups, up to finite factors. We also prove that G⁰ = G⁰⁰ when G is a definable subgroup of a linear algebraic group, over any model.

Cited by

• Weak regularity and one-parameter subgroups of definable p-adic Lie groups

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• Connected components of definable groups and o-minimality I
• Weak regularity and one-parameter subgroups of definable p-adic Lie groups
• Definably semisimple groups interpretable in p-adically closed fields