Kernel of the composition-factor homomorphism for a surface-group commensurator

Determine whether the kernel of the homomorphism d^{π₁(Σ)}: Comm(π₁(Σ)) → Z^(FS) is equal to AComm(π₁(Σ)) for the fundamental group π₁(Σ) of a closed surface Σ of genus at least two.

Background

The paper proves for a finitely generated free group F that the kernel of dF is precisely AComm(F), yielding a description of the monolith and of all proper quotients. The analogous equality for a closed surface group is unresolved, partly because the relevant automorphism actions on kernels of epimorphisms to finite simple groups can fail to be transitive.

References

The second one, more closely connected to the present work, is that we do not know whether the analogue of Theorem \ref{thm-ker-CommF} is true here, i.e. whether the kernel of $d{\pi_1(\Sigma)} : Comm(\pi_1(\Sigma)) \to Z{(FS)}$ is the subgroup $AComm(\pi_1(\Sigma))$.

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