Ivanov’s finite-index Frattini subgroup conjecture

Prove that for every finitely generated subgroup G of the mapping class group Mod(S_g) of a closed oriented surface of genus g≥3, the finite-index Frattini subgroup Φ_f(G) is nilpotent.

Background

The finite-index Frattini subgroup Φ_f(G) is the intersection of all finite-index maximal subgroups of a group G. Platonov’s theorem establishes nilpotency of this subgroup for finitely generated linear groups, motivating the analogous statement for subgroups of mapping class groups.

The paper proves the conjecture for finitely generated subgroups of the Torelli subgroup J_1(S_g), showing in fact that their finite-index Frattini subgroups are abelian. It also proves that the Andersen–Masbaum–Ueno conjecture would imply the full Ivanov conjecture, so the general assertion remains conditional in the paper.

References

This observation led Ivanov to formulate the following conjecture, which is the counterpart of Platonov's theorem for mapping class groups of surfaces: Let $G$ be a finitely generated subgroup of $\mathrm{Mod}(S_g)$ where $g\geq 3,$ then $\Phi_f(G)$ is nilpotent.

— Ivanov's finite-index Frattini subgroup conjecture for subgroups of the Torelli group  (2610.01734 - Detcherry, 1 Oct 2026) in Section 1, Introduction; Conjecture 1.1