Cutting and pasting in the Torelli subgroup of Out(Fn)

Landgraf, Jacob¹

¹Department of Mathematics, University of Notre Dame, Notre Dame, IN, United States

Algebraic and Geometric Topology, Tome 25 (2025) no. 1, pp. 1-38

Résumé

Using ideas from 3-manifolds, Hatcher–Wahl defined a notion of automorphism groups of free groups with boundary. We study their Torelli subgroups, adapting ideas introduced by Putman for surface mapping class groups. Our main results show that these groups are finitely generated, and also that they satisfy an appropriate version of the Birman exact sequence.

Détail

DOI : 10.2140/agt.2025.25.1

Keywords: automorphism groups of free groups, mapping class groups, Torelli group, $3$-manifolds, Birman exact sequence, finite generation

Bibliographie

[1] S Andreadakis, On the automorphisms of free groups and free nilpotent groups, Proc. Lond. Math. Soc. 15 (1965) 239 | DOI

[2] M A Armstrong, On the fundamental group of an orbit space, Proc. Cambridge Philos. Soc. 61 (1965) 639 | DOI

[3] S Bachmuth, Induced automorphisms of free groups and free metabelian groups, Trans. Amer. Math. Soc. 122 (1966) 1 | DOI

[4] J E Banks, The Birman exact sequence for 3-manifolds, Bull. Lond. Math. Soc. 49 (2017) 604 | DOI

[5] M Bestvina, K U Bux, D Margalit, Dimension of the Torelli group for Out(Fn), Invent. Math. 170 (2007) 1 | DOI

[6] T Brendle, N Broaddus, A Putman, The mapping class group of connect sums of S2 × S1, Trans. Amer. Math. Soc. 376 (2023) 2557 | DOI

[7] M R Bridson, K Vogtmann, Automorphism groups of free groups, surface groups and free abelian groups, from: "Problems on mapping class groups and related topics", Proc. Sympos. Pure Math. 74, Amer. Math. Soc. (2006) 301 | DOI

[8] T Church, B Farb, A Putman, Integrality in the Steinberg module and the top-dimensional cohomology of SLnOK, Amer. J. Math. 141 (2019) 1375 | DOI

[9] M Day, A Putman, A Birman exact sequence for Aut(Fn), Adv. Math. 231 (2012) 243 | DOI

[10] M Day, A Putman, The complex of partial bases for Fn and finite generation of the Torelli subgroup of Aut(Fn), Geom. Dedicata 164 (2013) 139 | DOI

[11] M Day, A Putman, A Birman exact sequence for the Torelli subgroup of Aut(Fn), Int. J. Algebra Comput. 26 (2016) 585 | DOI

[12] B Farb, D Margalit, A primer on mapping class groups, 49, Princeton Univ. Press (2012)

[13] A Hatcher, Homological stability for automorphism groups of free groups, Comment. Math. Helv. 70 (1995) 39 | DOI

[14] A Hatcher, N Wahl, Stabilization for the automorphisms of free groups with boundaries, Geom. Topol. 9 (2005) 1295 | DOI

[15] A Hatcher, N Wahl, Stabilization for mapping class groups of 3-manifolds, Duke Math. J. 155 (2010) 205 | DOI

[16] C A Jensen, N Wahl, Automorphisms of free groups with boundaries, Algebr. Geom. Topol. 4 (2004) 543 | DOI

[17] D Johnson, The structure of the Torelli group, III : The abelianization of T , Topology 24 (1985) 127 | DOI

[18] F Laudenbach, Sur les 2-sphères d’une variété de dimension 3, Ann. of Math. 97 (1973) 57 | DOI

[19] F Laudenbach, Topologie de la dimension trois: homotopie et isotopie, 12, Soc. Math. France (1974)

[20] H Maazen, Homology stability for the general linear group, PhD thesis, Utrecht University (1951)

[21] W Magnus, Über n-dimensionale Gittertransformationen, Acta Math. 64 (1935) 353 | DOI

[22] A Putman, Cutting and pasting in the Torelli group, Geom. Topol. 11 (2007) 829 | DOI

[23] A Putman, Obtaining presentations from group actions without making choices, Algebr. Geom. Topol. 11 (2011) 1737 | DOI

[24] A Putman, The commutator subgroups of free groups and surface groups, Enseign. Math. 68 (2022) 389 | DOI

[25] W Tomaszewski, A basis of Bachmuth type in the commutator subgroup of a free group, Canad. Math. Bull. 46 (2003) 299 | DOI

[26] N Wahl, From mapping class groups to automorphism groups of free groups, J. Lond. Math. Soc. 72 (2005) 510 | DOI