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

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 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.