Valued fields, metastable groups

We introduce a class of theories called metastable, including the theory of algebraically closed valued fields (ACVF) as a motivating example. The key local notion is of definable types dominated by their stable part. A theory is metastable (over a sort Γ) if every type over a sufficiently rich base structure can be viewed as part of a Γ-parametrized family of stably dominated types. We initiate a study of definable groups in metastable theories of finite rank. Groups with a stably dominated generic type are shown to have a canonical stable quotient. Abelian groups are shown to be decomposable into a part coming from Γ, and a definable direct limit system of groups with stably dominated generic. In the case of ACVF, among affine definable groups we characterize the groups with stably dominated