## Duke Mathematical Journal

### On finiteness properties of the Johnson filtrations

#### Abstract

Let $\Gamma$ be either the automorphism group of the free group of rank $n\geq4$ or the mapping class group of an orientable surface of genus $n\geq12$ with at most $1$ boundary component, and let $G$ be either the subgroup of $\mathrm{IA}$-automorphisms or the Torelli subgroup of $\Gamma$. For $N\in\mathbb{N}$ denote by $\gamma_{N}G$ the $N$th term of the lower central series of $G$. We prove that

(i) any subgroup of $G$ containing $\gamma_{2}G=[G,G]$ (in particular, the Johnson kernel in the mapping class group case) is finitely generated;

(ii) if $N=2$ or $n\geq8N-4$ and $K$ is any subgroup of $G$ containing $\gamma_{N}G$ (for instance, $K$ can be the $N$th 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 $\Gamma$ containing $\gamma_{N}G$, with $N$ as in (ii), then $H$ has finite Abelianization.

#### Article information

Source
Duke Math. J., Volume 167, Number 9 (2018), 1713-1759.

Dates
Revised: 14 January 2018
First available in Project Euclid: 3 May 2018

https://projecteuclid.org/euclid.dmj/1525313239

Digital Object Identifier
doi:10.1215/00127094-2018-0005

Mathematical Reviews number (MathSciNet)
MR3813595

Zentralblatt MATH identifier
06904638

#### Citation

Ershov, Mikhail; He, Sue. On finiteness properties of the Johnson filtrations. Duke Math. J. 167 (2018), no. 9, 1713--1759. doi:10.1215/00127094-2018-0005. https://projecteuclid.org/euclid.dmj/1525313239

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