ON GROUPS AND FIELDS DEFINABLE IN $1$-H-MINIMAL FIELDS

Abstract We show that an infinite group G definable in a $1$ -h-minimal field admits a strictly K -differentiable structure with respect to which G is a (weak) Lie group, and we show that definable local subgroups sharing the same Lie algebra have the same germ at the identity. We conclude that infinite fields definable in K are definably isomorphic to finite extensions of K and that $1$ -dimensional groups definable in K are finite-by-abelian-by-finite. Along the way, we develop the basic theory of definable weak K -manifolds and definable morphisms between them.