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

Abstract

Let K be the subgroup of SL _n (F[t]) consisting of matrices congruent to the identity modulo t. In [7], 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 (F[t]) are included as is the study of the analogous question for SL _n (F[t, t ⁻¹]).

ACKNOWLEDGMENT

Supported by NSF grant DMS-0070119.