Representation stability for filtrations of Torelli groups

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.

submitted version of my PhD thesis (http://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000104536); main theorems improved and theorems about noetherianness added

References in corpus (2)

• Cohomological Aspects of Magnus Expansions
• Central stability for the homology of congruence subgroups and the second homology of Torelli groups

Cited by in corpus (3)

• Effective and Infinite-Rank Superrigidity in the Context of Representation Stability
• Some generalizations of Schur functors
• Asymptotic behaviors of representations of graded categories with inductive functors