Homological stability for classical groups over finite fields

Abstract

if i < n and if k is any field with more than two elements. As discussed at this conference, no such stability theorem for other classical groups is known. In this paper, we give a prime-toresidue-characteristic homological stability theorem for classical groups over finite fields (Theorem 2), a stability theorem for the integral homology of classical groups over the algebraic closure of a finite field (Theorem 3), and a stability theorem for the integral homology of the special linear groups for finite fields with more than two elements (Theorem 6). We begin with the following well known stability results for the classical Lie groups. The proof is readily obtained by using , S 4n+3 ~ BSPn and the fibrations S 2n+l * BU n BUn+ 1 ~ BSPn+l, S n . BO n * BOn+ 1.