Borel's stable range for the cohomology of arithmetic groups

Abstract

In this note, we remark on the range in Borel's theorem on the stable cohomology of the arithmetic groups Sp(2n,Z) and SO(n,n;Z). This improves the range stated in Borel's original papers, an improvement that was known to Borel. Our main task is a technical computation involving the Weyl group action on roots and weights. This note originally appeared as the appendix to arXiv:1711.03139.