Duke Mathematical Journal

On finiteness properties of the Johnson filtrations

Mikhail Ershov and Sue He

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Let Γ be either the automorphism group of the free group of rank n4 or the mapping class group of an orientable surface of genus n12 with at most 1 boundary component, and let G be either the subgroup of IA-automorphisms or the Torelli subgroup of Γ. For NN denote by γNG the Nth term of the lower central series of G. We prove that

(i) any subgroup of G containing γ2G=[G,G] (in particular, the Johnson kernel in the mapping class group case) is finitely generated;

(ii) if N=2 or n8N4 and K is any subgroup of G containing γNG (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 Γ containing γNG, with N as in (ii), then H has finite Abelianization.

Article information

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

Received: 24 July 2017
Revised: 14 January 2018
First available in Project Euclid: 3 May 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 20F28: Automorphism groups of groups [See also 20E36]
Secondary: 20F40: Associated Lie structures 20J06: Cohomology of groups

finiteness properties Johnson filtration Johnson kernel automorphism group of a free group mapping class group Torelli subgroup


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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