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Representation stability for filtrations of Torelli groups

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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.

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Notes

  1. This issue can also be resolved by working with the Zariski closures \({{\mathrm{SL}}}_n^\pm \mathbb Q\) of \({{\mathrm{GL}}}_n \mathbb Z\) instead of \({{\mathrm{GL}}}_n\mathbb Q\) and an analogous category \(\mathsf {VIC}_\mathbb Q^\pm \) defined by Putman–Sam [29]. But this would make working with branching rules considerably harder.

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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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  1. Department of Mathematics, Purdue University, West Lafayette, IN, USA

    Peter Patzt

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  1. Peter Patzt
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Correspondence to Peter Patzt.

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Communicated by Andreas Thom.

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Patzt, P. Representation stability for filtrations of Torelli groups. Math. Ann. 372, 257–298 (2018). https://doi.org/10.1007/s00208-018-1708-6

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  • DOI: https://doi.org/10.1007/s00208-018-1708-6

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