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15 June 2018 On finiteness properties of the Johnson filtrations
Mikhail Ershov, Sue He
Duke Math. J. 167(9): 1713-1759 (15 June 2018). DOI: 10.1215/00127094-2018-0005


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.


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Mikhail Ershov. Sue He. "On finiteness properties of the Johnson filtrations." Duke Math. J. 167 (9) 1713 - 1759, 15 June 2018.


Received: 24 July 2017; Revised: 14 January 2018; Published: 15 June 2018
First available in Project Euclid: 3 May 2018

zbMATH: 06904638
MathSciNet: MR3813595
Digital Object Identifier: 10.1215/00127094-2018-0005

Primary: 20F28
Secondary: 20F40 , 20J06

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

Rights: Copyright © 2018 Duke University Press


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Vol.167 • No. 9 • 15 June 2018
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