On finiteness properties of the Johnson filtrations

Abstract

Let A denote either the automorphism group of the free group of rank n>=4 or the mapping class group of an orientable surface of genus n>=12 with at most 1 boundary component, and let G be either the subgroup of IA-automorphisms or the Torelli subgroup of A, respectively. For a natural number N denote by G_N the Nth term of the lower central series of G. We prove that (i) any subgroup of G containing [G,G] (in particular, the Johnson kernel in the mapping class group case) is finitely generated; (ii) if N=2 or n>=8N-4 and K is any subgroup of G containing G_N (for instance, K can be the Nth term of the Johnson filtration of G), then G/[K,K] is nilpotent and hence the abelianization of K is finitely generated; (iii) if H is any finite index subgroup of A containing G_N, with N as in (ii), then H has finite abelianization.