Ivanov's finite-index Frattini subgroup conjecture for subgroups of the Torelli group

Abstract: Let $S_g$ be a closed oriented surface of genus $g$. Ivanov conjectured that if $G$ a finitely generated subgroup of $\mathrm{Mod}(S_g)$ its finite-index Frattini subgroup $Φ_f(G)$ is nilpotent. We prove this conjecture for finitely generated subgroups of the Torelli subgroup $J_1(S_g)$, using recent partial results of the author about the Andersen-Masbaum-Ueno conjecture. We also show that the full AMU conjecture would imply Ivanov's conjecture.