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

TL;DR

This paper extends results on abelian $p$-adic definable groups to non-abelian groups, showing the existence of one-dimensional subgroups and the equality of certain connected components, with implications for definably amenable groups.

Contribution

It generalizes the Peterzil-Steinhorn theorem to non-abelian $p$-adic groups and proves $G^0=G^{00}$ for groups over $Q_p$ and linear algebraic groups.

Findings

Non-abelian definable groups not definably compact have one-dimensional non-compact subgroups.

For groups over $Q_p$, the connected components $G^0$ and $G^{00}$ coincide.

Definably amenable groups over $Q_p$ are essentially open subgroups of algebraic groups.

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 $\mathbb{Q}_p$, 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.