On finiteness properties of the Johnson filtrations

Mikhail Ershov, Sue He

Published 2017-03-12Version 1

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 1 boundary component, and let G be either the subgroup of IA-automorphisms or the Torelli subgroup of A, 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 the Nth term of the lower central series of G (for instance, if K is the Nth 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 A containing the Nth term of the lower central series of G, with N as above, then H has finite abelianization.