## Geometry & Topology

### Generating the Johnson filtration

#### Abstract

For $k ≥ 1$, let $ℐg1(k)$ be the term in the Johnson filtration of the mapping class group of a genus $g$ surface with one boundary component. We prove that for all $k ≥ 1$, there exists some $Gk ≥ 0$ such that $ℐg1(k)$ is generated by elements which are supported on subsurfaces whose genus is at most $Gk$. We also prove similar theorems for the Johnson filtration of $Aut(Fn)$ and for certain mod-$p$ analogues of the Johnson filtrations of both the mapping class group and of $Aut(Fn)$. The main tools used in the proofs are the related theories of FI–modules (due to the first author with Ellenberg and Farb) and central stability (due to the second author), both of which concern the representation theory of the symmetric groups over $ℤ$.

#### Article information

Source
Geom. Topol., Volume 19, Number 4 (2015), 2217-2255.

Dates
Revised: 21 August 2014
Accepted: 3 October 2014
First available in Project Euclid: 16 November 2017

https://projecteuclid.org/euclid.gt/1510858803

Digital Object Identifier
doi:10.2140/gt.2015.19.2217

Mathematical Reviews number (MathSciNet)
MR3375526

Zentralblatt MATH identifier
1364.20025

#### Citation

Church, Thomas; Putman, Andrew. Generating the Johnson filtration. Geom. Topol. 19 (2015), no. 4, 2217--2255. doi:10.2140/gt.2015.19.2217. https://projecteuclid.org/euclid.gt/1510858803

#### References

• H Bass, J Milnor, J-P Serre, Solution of the congruence subgroup problem for $\mathrm{ SL}\sb{n}(n\geq 3)$ and $\mathrm{Sp}\sb{2n}(n\geq 2)$, Inst. Hautes Études Sci. Publ. Math. (1967) 59–137
• M Bestvina, K-U Bux, D Margalit, Dimension of the Torelli group for $\mathrm{Out}(F\sb n)$, Invent. Math. 170 (2007) 1–32
• J S Birman, Mapping class groups and their relationship to braid groups, Comm. Pure Appl. Math. 22 (1969) 213–238
• N Bourbaki, Lie groups and Lie algebras, Chapters 1–3, Elements of Math. (Berlin), Springer (1989)
• T Church, J S Ellenberg, B Farb, FI–modules and stability for representations of symmetric groups, Duke Math. J. 164 (2015) 1833–1910
• T Church, J S Ellenberg, B Farb, R Nagpal, FI–modules over Noetherian rings, Geom. Topol. 18 (2014) 2951–2984
• J Cooper, Two ${\rm mod}$-$\!p$ Johnson filtrations, J. Topol. Anal. 7 (2015) 309–343
• M Day, A Putman, The complex of partial bases for $F\sb n$ and finite generation of the Torelli subgroup of $\mathrm{Aut}(F\sb n)$, Geom. Dedicata 164 (2013) 139–153
• N Enomoto, T Satoh, On the derivation algebra of the free Lie algebra and trace maps, Algebr. Geom. Topol. 11 (2011) 2861–2901
• E Fadell, L Neuwirth, Configuration spaces, Math. Scand. 10 (1962) 111–118
• B Farb, D Margalit, A primer on mapping class groups, Princeton Math. Series 49, Princeton Univ. Press (2012)
• S Garoufalidis, J Levine, Finite type $3$–manifold invariants and the structure of the Torelli group, I, Invent. Math. 131 (1998) 541–594
• A Hatcher, D Margalit, Generating the Torelli group, Enseign. Math. 58 (2012) 165–188
• N V Ivanov, J D McCarthy, On injective homomorphisms between Teichmüller modular groups, I, Invent. Math. 135 (1999) 425–486
• D L Johnson, Homeomorphisms of a surface which act trivially on homology, Proc. Amer. Math. Soc. 75 (1979) 119–125
• D L Johnson, An abelian quotient of the mapping class group $\mathcal{ I}\sb{g}$, Math. Ann. 249 (1980) 225–242
• D L Johnson, The structure of the Torelli group, I: A finite set of generators for $\mathcal{I}$, Ann. of Math. 118 (1983) 423–442
• D L Johnson, A survey of the Torelli group, from: “Low-dimensional topology”, (S J Lomonaco, Jr, editor), Contemp. Math. 20, Amer. Math. Soc. (1983) 165–179
• D L Johnson, The structure of the Torelli group, II: A characterization of the group generated by twists on bounding curves, Topology 24 (1985) 113–126
• D L Johnson, The structure of the Torelli group, III: The abelianization of $\mathcal{T}$, Topology 24 (1985) 127–144
• W B R Lickorish, A finite set of generators for the homeotopy group of a $2$–manifold, Proc. Cambridge Philos. Soc. 60 (1964) 769–778
• W Magnus, Über $n$–dimensionale Gittertransformationen, Acta Math. 64 (1935) 353–367
• M Matsumoto, Introduction to arithmetic mapping class groups, from: “Moduli spaces of Riemann surfaces”, (B Farb, R Hain, E Looijenga, editors), IAS/Park City Math. Ser. 20, Amer. Math. Soc. (2013) 319–356
• D Mumford, Abelian quotients of the Teichmüller modular group, J. Analyse Math. 18 (1967) 227–244
• M Newman, J R Smart, Symplectic modulary groups, Acta Arith 9 (1964) 83–89
• B Perron, Filtration de Johnson et groupe de Torelli modulo $p$, $p$ premier, C. R. Math. Acad. Sci. Paris 346 (2008) 667–670
• J Powell, Two theorems on the mapping class group of a surface, Proc. Amer. Math. Soc. 68 (1978) 347–350
• A Putman, Cutting and pasting in the Torelli group, Geom. Topol. 11 (2007) 829–865
• A Putman, Small generating sets for the Torelli group, Geom. Topol. 16 (2012) 111–125
• A Putman, Stability in the homology of congruence subgroups, Invent. Math. (2015) Published online in March 2015
• T Satoh, On the Johnson filtration of the basis-conjugating automorphism group of a free group, Michigan Math. J. 61 (2012) 87–105
• T Satoh, On the lower central series of the $\mathrm{IA}$–automorphism group of a free group, J. Pure Appl. Algebra 216 (2012) 709–717
• J Stallings, Homology and central series of groups, J. Algebra 2 (1965) 170–181
• B Sury, T N Venkataramana, Generators for all principal congruence subgroups of $\mathrm{ SL}(n,\mathbb{Z})$ with $n\geq 3$, Proc. Amer. Math. Soc. 122 (1994) 355–358
• J Tits, Systèmes générateurs de groupes de congruence, C. R. Acad. Sci. Paris Sér. A-B 283 (1976) A693–A695
• E Witt, Treue Darstellungen Liescher Ringe, J. Reine Angew. Math. 177 (1937) 152–160
• H Zassenhaus, Ein Verfahren, jeder endlichenp–Gruppe einen Lie–Ring mit der Charakteristikp zuzuordnen, Abh. Math. Sem. Univ. Hamburg 13 (1939) 200–207