Abelianity of proper quotients of the commensurator of a closed surface group

Determine whether every proper quotient of the abstract commensurator Comm(π₁(Σ)) of the fundamental group of a closed surface Σ of genus at least two is abelian.

Background

The paper proves that every proper quotient of the abstract commensurator of a finitely generated free group is abelian. It establishes an analogous description of the image of the determinant-like homomorphism d{π₁(Σ)} for a closed surface group, but leaves unresolved whether the corresponding conclusion about all proper quotients holds for surface groups.

References

For instance we do not know whether the analogue of Corollary \ref{cor-ker-CommF-monolith} is true, i.e. whether every proper quotient of $Comm(\pi_1(\Sigma))$ is abelian.

— Free abelian quotients of commensurators  (2609.35323 - Boudec, 28 Sep 2026) in Section 3, subsection “On the abstract commensurator of surface groups”