Nonvanishing of group cohomology of SL(n,ℤ)

Abstract

It has been recently shown by Bader and Sauer that H^k(SL(n,ℤ);π) = 0 for k <= n − 2 and π being an orthogonal representation without non-zero invariant vectors. We show for n = 3 and n = 4 the existence of orthogonal representations π of SL(n,ℤ) which do not have nontrivial invariant vectors and for which H^{n−1}(SL(n,ℤ);π) is nonzero. This proves that the Bader-Sauer result is sharp for these degrees. This is the joint work with B. Br ̈uck, S. Hughes and D. Kielak.