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Mathematische Annalen

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

Representation stability for filtrations of Torelli groups

  • Peter Patzt
Article

Abstract

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 

Notes

Acknowledgements

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.

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Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsPurdue UniversityWest LafayetteUSA

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