Ivanov's finite-index Frattini conjecture for subgroups of mapping class groups

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 ${\Phi }_f(G)$ is nilpotent. We prove this conjecture, using recent partial results of the first author about the Andersen-Masbaum-Ueno conjecture.

Corrected proof of Lemma 2.1. Added new group-theoretic Lemma 3.2. The main result is now unconditional, while the proof is shorter !