Submission date: 3 August 2023

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 G^0 = G^{00}. As an application, definably amenable groups over 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.

Mathematics Subject Classification: 03C60

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