Let be either the automorphism group of the free group of rank or the mapping class group of an orientable surface of genus with at most boundary component, and let be either the subgroup of -automorphisms or the Torelli subgroup of . For denote by the th term of the lower central series of . We prove that
(i) any subgroup of containing (in particular, the Johnson kernel in the mapping class group case) is finitely generated;
(ii) if or and is any subgroup of containing (for instance, can be the th term of the Johnson filtration of ), then is nilpotent and hence the Abelianization of is finitely generated;
(iii) if is any finite-index subgroup of containing , with as in (ii), then has finite Abelianization.
Mikhail Ershov. Sue He. "On finiteness properties of the Johnson filtrations." Duke Math. J. 167 (9) 1713 - 1759, 15 June 2018. https://doi.org/10.1215/00127094-2018-0005