Let Gamma_{0,n}(p) be the congruence subgroup of level p of SL_n(Z) whose first column is congruent to (*,0,\dots,0)^t \mod p. The cohomology of this subgroup has connections to problems in algebraic K-theory and number theory. Borel and Serre (1973) showed that the rational cohomology of Gamma_{0,n}(p) vanishes above degree n(n+1)/2.

We prove that the top-dimensional rational cohomology group of Gamma_{0,n}(p) vanishes for all p equal to 2,3,5,7,13 and when n is at least 3, as well as for all primes p at most 6n-14. We also reprove the known non-vanishing result that this group is nonzero for n=2 for every prime p, and we establish a new non-vanishing for n=3 for all primes p not equal to 2,3,5,7,13.

In this talk, I will outline the ideas behind these results and briefly survey what is known about the top-dimensional cohomology of related congruence subgroups.