CONGRUENCE SUBGROUPS AND TWISTED COHOMOLOGY OF SLn (F[t]). II. FINITE FIELDS AND NUMBER FIELDS

Let K be the subgroup of SL _n (Ft]) consisting of matrices congruent to the identity modulo t. In 7] Knudson, K. 1998. Congruence Subgroups and Twisted Cohomology of SL_n(Ft]). J. Algebra, 207: 695–721.  Google Scholar], the author conjectured that if F is a finite field, then H ₁(K) is the adjoint representation s l _n (F). A proof of this conjecture is provided in this note. The argument also works in case F is a number field. Applications to the cohomology of SL _n (Ft]) are included as is the study of the analogous question for SL _n (Ft, t ^?1]).