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arXiv · 1703.04190

On finiteness properties of the Johnson filtrations

Mikhail Ershov·Sue He
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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.

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Mikhail Ershov, Sue He. 2018-03-12. On finiteness properties of the Johnson filtrations. https://doi.org/10.1215/00127094-2018-0005

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