Mathematische Annalen

, Volume 372, Issue 1–2, pp 257–298 | Cite as

Representation stability for filtrations of Torelli groups

  • Peter Patzt


We show, finitely generated rational \(\mathsf {VIC}_\mathbb Q\)-modules and \(\mathsf {SI}_\mathbb Q\)-modules are uniformly representation stable and all their submodules are finitely generated. We use this to prove two conjectures of Church and Farb, which state that the quotients of the lower central series of the Torelli subgroups of \({{\mathrm{Aut}}}(F_n)\) and \({{\mathrm{Mod\,}}}(\Sigma _{g,1})\) are uniformly representation stable as sequences of representations of the general linear groups and the symplectic groups, respectively. Furthermore we prove an analogous statement for their Johnson filtrations.

Mathematics Subject Classification

Primary 20G05 Secondary 20E36 20F40 58D05 



First and foremost the author wishes to thank his advisor Holger Reich for introducing him to the interesting and emerging field of representation stability. During his PhD the author was supported by the Berlin Mathematical School, the SFB Raum–Zeit–Materie and the Dahlem Research School. The author also wants to thank Kevin Casto, Tom Church, Daniela Egas Santander, Benson Farb, Daniel Lütgehetmann, Jeremy Miller, Holger Reich, Steven Sam, David Speyer and Elmar Vogt for helpful conversations. Special thanks to Steven Sam for his extensive help with the modification rules, and to Kevin Casto for pointing out the conjectures to the author. The author would also like to thank the anonymous referee that suggested several improvements to the current version.


  1. 1.
    Andreadakis, S.: On the automorphisms of free groups and free nilpotent groups. Proc. Lond. Math. Soc. 3(15), 239–268 (1965)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bachmuth, S.: Induced automorphisms of free groups and free metabelian groups. Trans. Am. Math. Soc. 122, 1–17 (1966)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Boldsen, S.K., Dollerup, M.H.: Towards representation stability for the second homology of the Torelli group. Geom. Topol. 16(3), 1725–1765 (2012)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Borel, A.: Density properties for certain subgroups of semi-simple groups without compact components. Ann. Math. 2(72), 179–188 (1960)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Bourbaki, N.: Lie groups and Lie algebras. Chapters 1–3. In: Elements of Mathematics (Berlin). Springer, Berlin (1998). Translated from the French, Reprint of the 1989 English translationGoogle Scholar
  6. 6.
    Church, T., Ellenberg, J.S., Farb, B.: FI-modules and stability for representations of symmetric groups. Duke Math. J. 164(9), 1833–1910 (2015)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Church, T., Farb, B.: Representation theory and homological stability. Adv. Math. 245, 250–314 (2013)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Church, T., Putman, A.: Generating the Johnson filtration. Geom. Topol. 19(4), 2217–2255 (2015)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Day, M., Putman, A.: On the second homology group of the Torelli subgroup of \({\rm {Aut}}(F_n)\). Geom. Topol. 21(5), 2851–2896 (2017)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Djament, A.: Des propriétés de finitude des foncteurs polynomiaux. Fundam. Math. 233(3), 197–256 (2016)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Formanek, E.: Characterizing a free group in its automorphism group. J. Algebra 133(2), 424–432 (1990)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Fulton, W., Harris, J.: Representation theory. In: Graduate Texts in Mathematics, vol. 129. Springer, New York (1991). A first course, Readings in MathematicsGoogle Scholar
  13. 13.
    Fulton, W.: Young tableaux. In: London Mathematical Society Student Texts, vol. 35. Cambridge University Press, Cambridge, With applications to representation theory and geometry (1997)Google Scholar
  14. 14.
    Gan, W.L., Watterlond, J.: Stable decompositions of certain representations of the finite general linear groups. Transform. Groups 23(2), 425–435 (2018)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Goodman, R., Wallach, N.R.: Symmetry, representations, and invariants. In: Graduate Texts in Mathematics, vol. 255. Springer, Dordrecht (2009)Google Scholar
  16. 16.
    Green, J.A.: Polynomial representations of \({\rm GL}\_{n}\). In: Lecture Notes in Mathematics, vol. 830. Springer, Berlin, augmented edition (2007). With an appendix on Schensted correspondence and Littelmann paths by K. Erdmann, Green and M. SchockerGoogle Scholar
  17. 17.
    Hain, R.: Infinitesimal presentations of the Torelli groups. J. Am. Math. Soc. 10(3), 597–651 (1997)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hönig, C.S.: Proof of the well-ordering of cardinal numbers. Proc. Am. Math. Soc. 5, 312 (1954)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Habegger, N., Sorger, C.: An infinitesimal presentation of the Torelli group of a surface with boundary. Preprint, 2000.
  20. 20.
    Howe, R., Tan, E.-C., Willenbring, J.F.: Stable branching rules for classical symmetric pairs. Trans. Am. Math. Soc. 357(4), 1601–1626 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Jantzen, J.C.: Representations of algebraic groups. In: Pure and Applied Mathematics, vol. 131. Academic Press Inc, Boston (1987)Google Scholar
  22. 22.
    Johnson, D.: An abelian quotient of the mapping class group \({\cal{I}}_{g}\). Math. Ann. 249(3), 225–242 (1980)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Johnson, D.: The structure of the Torelli group. III. The abelianization of \(\mathscr {I}\). Topology 24(2), 127–144 (1985)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Koike, K.: On the decomposition of tensor products of the representations of the classical groups: by means of the universal characters. Adv. Math. 74(1), 57–86 (1989)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Koike, K., Terada, I.: Young-diagrammatic methods for the representation theory of the classical groups of type \(B_n,\, C_n,\, D_n\). J. Algebra 107(2), 466–511 (1987)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lazard, M.: Sur les groupes nilpotents et les anneaux de Lie. Ann. Sci. Ecole Norm. Sup. 3(71), 101–190 (1954)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Magnus, W.: über Gruppen und zugeordnete Liesche Ringe. J. Reine Angew. Math. 182, 142–149 (1940)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Miller, J., Patzt, P., Wilson, J.C.H.: Central stability for the homology of congruence subgroups and the second homology of Torelli groups. Preprint, 04 (2017). arXiv:1704.04449
  29. 29.
    Putman, A., Sam, S.V.: Representation stability and finite linear groups. Duke Math. J. 166(13), 2521–2598 (2017)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Randal-Williams, O., Wahl, N.: Homological stability for automorphism groups. Adv. Math. 318, 534–626 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Sam, S.V., Snowden, A., Weyman, J.: Homology of Littlewood complexes. Selecta Math. (N.S.) 19(3), 655–698 (2013)MathSciNetCrossRefGoogle Scholar
  32. 32.
    Satoh, T.: On the lower central series of the IA-automorphism group of a free group. J. Pure Appl. Algebra 216(3), 709–717 (2012)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Satoh, T.: A survey of the Johnson homomorphisms of the automorphism groups of free groups and related topics. In: Handbook of Teichmüller theory. vol. V, pp. 167–209. European Mathematical Society (EMS), Zürich (2016)Google Scholar
  34. 34.
    Serre, J.-P.: Lie algebras and Lie groups. In: Lecture Notes in Mathematics. Springer, Berlin (2006). 1964 lectures given at Harvard University, Corrected fifth printing of the second (1992) editionGoogle Scholar
  35. 35.
    Snowden, A.: Syzygies of Segre embeddings and \(\Delta \)-modules. Duke Math. J. 162(2), 225–277 (2013)MathSciNetCrossRefGoogle Scholar
  36. 36.
    Weyl, H.: The Classical Groups. Their Invariants and Representations. Princeton University Press, Princeton (1939)zbMATHGoogle Scholar
  37. 37.
    Witt, E.: Treue Darstellung Liescher Ringe. J. Reine Angew. Math. 177, 152–160 (1937)MathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

Personalised recommendations