On Finiteness Properties of the Johnson Filtrations

Let Γ 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 1 boundary component, and let G be either the subgroup of IA-automorphisms or the Torelli subgroup of Γ, respectively. We prove that any subgroup of G containing [G,G] (in particular, the Johnson kernel in the mapping class group case) is finitely generated. We also prove that if N ≤ 1 + n 12 and K is any subgroup of G containing γNG, the N th term of the lower central series of G (for instance, if K is the N th term of the Johnson filtration of G), then the abelianization K/[K,K] is finitely generated. Finally, we prove that if H is any finite index subgroup of Γ containing γNG, then H has finite abelianization.