Journal of Topology Advance Access originally published online on December 9, 2007
Journal of Topology 2008 1(2):306-316; doi:10.1112/jtopol/jtm002
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© 2007 London Mathematical Society
The integral Novikov conjectures for linear groups containing torsion elements
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA
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Abstract |
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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 
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 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 is positive, many of these S-arithmetic subgroups such as 
do not admit cofinite universal spaces for proper actions.
Received February 15, 2007.
2000 Mathematics Subject Classification 57R67, 57R19, 19D50 (primary)
This work was partially supported by NSF grants DMS 0405884 and DMS 0604878.