Central stability for the homology of congruence subgroups and the second homology of Torelli groups

Abstract

We prove a representation stability result for the second homology groups of Torelli subgroups of mapping class groups and automorphism groups of free groups. This strengthens the results of Boldsen-Hauge Dollerup and Day-Putman. We also prove a new representation stability result for the homology of certain congruence subgroups, partially improving upon the work of Putman-Sam. These results follow from a general theorem on syzygies of certain modules with finite polynomial degree.

Corrections and substantial revisions. 36 pages, 1 figure. To appear in Advances in Mathematics

References in corpus (7)

• Cohomological Aspects of Magnus Expansions
• Dimension of the Torelli group for Out(F_n)
• A long exact sequence for homology of FI-modules
• On the cohomology of Torelli groups
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Cited by in corpus (11)

• On the cohomology of Torelli groups
• Stability in the high-dimensional cohomology of congruence subgroups
• Representation stability for filtrations of Torelli groups
• Representation stability, secondary stability, and polynomial functors
• Linear stable range for homology of congruence subgroups via FI-modules
• Effective and Infinite-Rank Superrigidity in the Context of Representation Stability
• Equivariant group presentations and the second homology group of the Torelli group
• On spectral sequence for the action of genus 3 Torelli group on the complex of cycles
• Polynomial conditions and homology of FI-modules
• Asymptotic behaviors of representations of graded categories with inductive functors
• Partial Torelli groups and homological stability