Representation stability in the level 4 braid group

We investigate the cohomology of level 4 subgroup braid group, namely, kernel mod reduction Burau representation at $$t=-1$$ . This group is also equal to 2 abelianization pure group. give an exact formula for first Betti number; it a quartic polynomial in number strands. show that, like homology satisfies uniform stability sense Church and Farb. Unlike symmetries—the quotient by subgroup—is one which theory has not been well studied; we develop its theory. non-split extension symmetric As applications our main results, that rational ring generated degree 1 when strands least 15, compute all numbers most 4. derive new lower bound on hyperelliptic Torelli top mapping class genus 2. Finally, apply results locate 2-torsion points characteristic varieties