Bounded orbits and strongly generic sets

Let G be a group definable in a theory T. We study the action of G on the space of its external types. We introduce the notion of a strongly generic subset of G. We prove that there is a bounded orbit of external types if and only if there are boundedly many externally definable strongly generic subsets of G. In this situation, we prove the following. The group G00 exists and equals G∞, also the size of every bounded orbit of an external almost periodic type is bounded by 2ℵ0. The ideal subgroups of the space of external types of G up to isomorphism do not depend on the choice of a model of T. If additionally T is weakly o‐minimal, then G is definably amenable.