Generic transitivity of idempotent generically stable types

Determine whether every idempotent generically stable global type p in S_G(U), for a type‑definable group G in an arbitrary first‑order structure, is generically transitive; equivalently, show that for any small model M over which p is invariant and any realization (a1,a0) of p^{(2)}, the pair (a1·a0,a0) also realizes p^{(2)} (equivalently, p ∈ S_{Stab_ℓ(p)}(U)).

Background

The paper introduces the notion of generic transitivity for a global type p ∈ S_G(U) and shows it is equivalent to p lying on its left (or right) stabilizer subgroup. This property is proved in several settings (stable theories, abelian groups, inp‑minimal groups, and rosy theories), but the general case is unresolved.

The authors highlight that establishing generic transitivity for all idempotent generically stable types would complete a key step in extending stable group theory phenomena beyond stable and rosy contexts.