  We introduce a technique for proving quantitative representation stability
theorems for sequences of representations of certain finite linear groups over
a field of characteristic zero. In particular, we prove a vanishing result for
higher syzygies of VIC- and SI-modules, which can be thought of as a weaker
version of a regularity theorem of Church-Ellenberg in the context of
FI-modules. We apply these techniques to the rational homology of congruence
subgroups of mapping class groups and congruence subgroups of automorphism
groups of free groups. This partially resolves a question raised by Church and
Putman-Sam. We also prove new homological stability results for mapping class
groups and automorphism groups of free groups with twisted coefficients.
