The integral Novikov conjectures for linear groups containing torsion elements

In this paper, we show that for any global field k, the generalizedintegral Novikov conjecture in both K- and L-theories holdsfor every finitely generated subgroup of GL(n, k). This impliesthat the conjecture holds for every finitely generated subgroupof , where is the algebraic closure of . We also show that for every linear algebraicgroup defined over k, every S-arithmetic subgroup satisfiesthis generalized integral Novikov conjecture. We note that theintegral Novikov conjecture implies the stable Borel conjecture,in particular, the stable Borel conjecture holds for all theabove torsion-free groups. Most of these subgroups are not discretesubgroups of Lie groups with finitely many connected components,and some of them are not finitely generated. When the fieldk is a function field such as , and the k-rank of is positive, many of these S-arithmeticsubgroups such as donot admit cofinite universal spaces for proper actions. Received February 15, 2007.