Abstract We construct explicit finite-dimensional orthogonal representations N of SL₍ (Z) for N \3, 4\ all of whose invariant vectors are trivial, and such that H^N - 1 (SL₍ (Z), N) is non-trivial. This implies that for N as above, the group SL₍ (Z) does not have property (T₍-₁) of Bader–Sauer and therefore is not (N-1) -Kazhdan in the sense of De Chiffre–Glebsky–Lubotzky–Thom, both being higher versions of Kazhdan’s property T.
Brück et al. (Fri,) studied this question.