Definability of stabilizers: intersection of definable groups

Establish whether, for a type‑definable group G and a generically stable idempotent type p ∈ S_G(U), the left stabilizer Stab_ℓ(p) is an intersection of M‑definable subgroups of G(U) in general (beyond the definable G case or the generically transitive assumption).

Background

The paper proves that if p is generically transitive, then Stab_ℓ(p) is an intersection of definable groups. For definable G this is immediate, but for type‑definable G the general case remains unresolved.

Understanding definability properties of stabilizers is key to transferring stable group techniques to broader settings and to describing invariant measures and components.