Groups and pregeometries

Question

Definition. For an infinite structure $\mathcal{A}$ and $cl : P(dom(\mathcal{A})) \longrightarrow P(dom(\mathcal{A}))$ , we say that $(\mathcal{A}, cl)$ is a structure carrying an $\omega$-homogeneous pregeometry if the following holds:

(a) $(\mathcal{A}, cl)$ is a pregeometry,

(b) $dim(\mathcal{A})$ is the same as the cardinality of $\mathcal{A}$,

(c) If $A \subseteq \mathcal{A}$ is finite and $a, b \in \mathcal{A} \setminus cl(A)$ , then there is $f\in Aut(\mathcal{A}/A)$ such that $f(a) = b$ and for all $B \subseteq \mathcal{A}$, $f(cl(B)) = cl(f(B))$.

Theorem. Suppose that $(G, cl)$ is a group carrying an $\omega$-homogeneous pregeometry. Then either $G$ is commutative or unstable.

Question. Are there non-commutative groups that carry an $\omega$-homogeneous pregeometry?

Giving references is appreciated.

Answer 1

This question has been around for some time.

Connections of homogeneous pregeomtries, quasiminimal structures and regular types have been studied in a recent article of Pilay and Tanovic. They show that the generic type of a homogeneous pregeometry is strongly regular (and generically stable) and conversely a global strongly regular type induces a homogeneous pregeomtry (Theorem 3). Earlier they analyse regular groups and ask if every regular group is commutative (which a variant of your question), see the question after Theorem 2.

As far as I know, the question is still open.