# On finiteness properties of the Johnson filtrations

## TL;DR

This paper investigates the finiteness properties of Johnson filtrations in automorphism groups of free groups and mapping class groups, proving finite generation and nilpotency results for certain subgroups and their quotients.

## Contribution

It establishes new finiteness and nilpotency properties of subgroups within Johnson filtrations, extending understanding of their algebraic structure.

## Key findings

- Subgroups containing the commutator subgroup are finitely generated.
- Certain quotients of these groups are nilpotent.
- Finite index subgroups have finite abelianization.

## 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.