The integral Novikov conjectures for linear groups containing torsion elements

Abstract

In this paper, we show that for any global field k, the generalized integral Novikov conjecture in both K- and L-theories holds for every finitely generated subgroup Γ of GL(n, k). This implies that the conjecture holds for every finitely generated subgroup of , where is the algebraic closure of . We also show that for every linear algebraic group Γ defined over k, every S-arithmetic subgroup satisfies this generalized integral Novikov conjecture. We note that the integral Novikov conjecture implies the stable Borel conjecture, in particular, the stable Borel conjecture holds for all the above torsion-free groups. Most of these subgroups are not discrete subgroups of Lie groups with finitely many connected components, and some of them are not finitely generated. When the field k is a function field such as , and the k-rank of Γ is positive, many of these S-arithmetic subgroups such as do not admit cofinite universal spaces for proper actions.