Geodesic
Effective finite generation for $[\mathrm{ IA}_n,\mathrm{ IA}_n]$ and the Johnson kernel
Mikhail Ershov1 ; Daniel Franz2
1University of Virginia, Charlottesville, USA
2Jacksonville University, Jacksonville, USA
Groups, geometry, and dynamics, Tome 17 (2023) no. 4, pp. 1149-1192

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DOI

Let IAn​ denote the group of IA-automorphisms of a free group of rank n, and let Inb​ denote the Torelli subgroup of the mapping class group of an orientable surface of genus n with b boundary components, b=0,1. In 1935, Magnus proved that IAn​ is finitely generated for all n, and in 1983, Johnson proved that Inb​ is finitely generated for n≥3. It was recently shown that for each k∈N, the k-th terms of the lower central series γk​IAn​ and γk​Inb​ are finitely generated when n≫k; however, no information about finite generating sets was known for k>1. The main goal of this paper is to construct an explicit finite generating set for γ2​IAn​=[IAn​,IAn​] and almost explicit finite generating sets for γ2​Inb​ and the Johnson kernel, which contains γ2​Inb​ as a finite index subgroup.
DOI : 10.4171/ggd/727
Classification : 20-XX, 57-XX
Mots-clés : Automorphism groups of free groups, mapping class groups, Torelli subgroup, Johnson kernel, BNS invariant

Mikhail Ershov  1   ; Daniel Franz  2

1 University of Virginia, Charlottesville, USA
2 Jacksonville University, Jacksonville, USA 
Mikhail Ershov; Daniel Franz. Effective finite generation for $[\mathrm{ IA}_n,\mathrm{ IA}_n]$ and the Johnson kernel. Groups, geometry, and dynamics, Tome 17 (2023) no. 4, pp. 1149-1192. doi: 10.4171/ggd/727
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     title = {Effective finite generation for $[\mathrm{ IA}_n,\mathrm{ IA}_n]$ and the {Johnson} kernel},
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