# High dimensional cohomology of congruence subgroups

# High dimensional cohomology of congruence subgroups

The level p congruence subgroup of SL_n(Z) is defined to be the subgroup of matrices congruent to the identity matrix mod p. These groups have trivial cohomology in high enough degrees. In the 1970s, Lee and Szczarba gave a conjectural description of the top dimensional cohomology groups of these congruence subgroups. In joint work with Miller and Putman, we resolve this conjecture by proving it for p=2,3,5 (p=3 was known) and disproving it for larger primes by finding more cohomology than conjectured. In particular, we compute the top dimensional cohomology of these groups for p=2,3,5 and we find new exotic cohomology classes for p at least 7 coming from the first homology group of the associated compactified modular curve.

I will also discuss joint work with Miller and Nagpal on a stability pattern in the high dimensional cohomology of congruence subgroups.