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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

Lizhen Ji

Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA

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 {Gamma} of GL(n, k). This implies that the conjecture holds for every finitely generated subgroup of Formula , where Formula is the algebraic closure of Formula . We also show that for every linear algebraic group {Gamma} 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 Formula , and the k-rank of {Gamma} is positive, many of these S-arithmetic subgroups such as Formula 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.



References

  1. Bartels A. ‘Squeezing and higher algebraic K-theory’. K-Theory (2003) 28:19–37.
  2. Bartels A., Rosenthal D. ‘On the K-theory of groups with finite asymptotic dimension’. Preprint, 2006 arXiv:math.KT/0605088.
  3. Behr H. ‘Higher finiteness properties of S-arithmetic groups in the function field case I’. Groups: Topological, combinatorial and arithmetic aspects. London Mathematical Society Lecture Note Series, 311. (Cambridge University Press, Cambridge, 2004).
  4. Behr H. ‘Arithmetic groups over function fields. I. A complete characterization of finitely generated and finitely presented arithmetic subgroups of reductive algebraic groups’. J. Reine Angew. Math. (1998) 495:79–118.
  5. Bell G., Dranishnikov A. ‘A Hurewicz-type theorem for asymptotic dimension and applications to geometric group theory’. Trans. Amer. Math. Soc. (2006) 358:4749–4764.[CrossRef]
  6. Bökstedt M., Hsiang W., Madsen I. ‘The cyclotomic trace and algebraic K-theory of spaces’. Invent. Math. (1993) 111:465–539.[CrossRef]
  7. Borel A., Serre J. P. ‘Corners and arithmetic groups’. Comment. Math. Helv. (1973) 48:436–491.[CrossRef]
  8. Borel A., Serre J. P. ‘Cohomologie dímmeubles et de groupes S-arithmétiques’. Topology (1976) 15:211–232.[CrossRef][ISI]
  9. Bridson M., Haefliger A. Metric spaces of non-positive curvature. Grundlehren der Mathematischen Wissenschaften} 319 (Springer, Berlin, 1999).
  10. Brown K. ‘Groups of virtually finite dimension’. Homological group theory. 27–70. London Mathematical Society Lecture Note Series 36. (Cambridge University Press, Cambridge, 1979).
  11. Brown K. Cohomology of groups. (Springer, Berlin, 1982).
  12. Brown K. ‘Finiteness properties of groups’. J. Pure Appl. Algebra (1987) 44:45–75.[CrossRef]
  13. Brown K. Buildings. (Springer, Berlin, 1988).
  14. Bux K., Wortman K. ‘A geometric proof that SL2(Z[t,t–1]) is not finitely presented’. Algebraic Geom. Topol. (2006) 6:839–852.[CrossRef]
  15. Bux K., Wortman K. ‘Finiteness properties of arithmetic groups over function fields’. Invent. Math. (2007) 167:355–378.[CrossRef]
  16. Carlsson G., Pedersen E. ‘Controlled algebra and the Novikov conjectures for K- and L-theory’. Topology (1995) 34:731–758.[CrossRef][ISI]
  17. Carlsson G., Goldfarb B. ‘The integral K-theoretic Novikov conjecture for groups with finite asymptotic dimension’. Invent. Math. (2004) 157:405–418.
  18. Carlsson G., Goldfarb B. ‘On homological coherence of discrete groups’. J. Algebra (2004) 276:502–514.[CrossRef]
  19. Carter D. ‘Localization in lower algebraic K-theory’. Commun. Algebra (1980) 8:603–622.[CrossRef]
  20. Chang S., Ferry S., Yu G. ‘Bounded rigidity of manifolds and asymptotic dimension growth’. K-theory (2007) in press.
  21. Dranishnikov A., Ferry S., Weinberger S. ‘An etale approach to the Novikov conjecture’. Preprint 2005, math.GT/0509644.
  22. Eberlein P. Geometry of nonpositively curved manifolds. Chicago Lectures in Mathematics (University of Chicago Press, Chicago, 1996).
  23. Farrell F., Hsiang W. C. ‘On Novikov's conjecture for nonpositively curved manifolds. I’’. Ann. Math. (1981) 113:199–209.[CrossRef]
  24. Farrell F., Lafont J. ‘EZ-structures and topological applications’. Comment. Math. Helv. (2005) 80:103–121.
  25. , Farrell F., Wagoner J. ‘Algebraic torsion for infinite simple homotopy types’. Comment. Math. Helv. (1972) 47:502–513.[CrossRef]
  26. Gromov M. ‘Asymptotic invariants of infinite groups’’. Geometric group theory—Niblo A., Roller M., eds. Cambridge University Press, Cambridge).
  27. Guentner E., Higson N., Weinberger S. ‘The Novikov conjecture for linear groups’. Math. Publ. d'IHES (2005) 101:243–268.[CrossRef]
  28. Illman S. ‘Existence and uniqueness of equivariant triangulations of smooth proper G-manifolds with some applications to equivariant Whitehead torsion’. J. Reine Angew. Math. (2000) 524:129–183.
  29. Ji L. ‘Asymptotic dimension and the integral K-theoretic Novikov conjecture for arithmetic groups’. J. Differential Geom. (2004) 68:535–544.
  30. Ji L. ‘Large scale geometry, compactifications and the integral Novikov conjectures for arithmetic groups’. Proceedings of the International Congress of Chinese Mathematicians, 2004, in press.
  31. Ji L. ‘Integral Novikov conjectures and arithmetic groups containing torsion elements’. Comm. Anal. Geom. (2007) 15:99–123.
  32. Ji L. ‘Integral Novikov conjectures for S-arithmetic groups I’. K-theory (2007) 38:35–47.
  33. Ji L. ‘Integral Novikov conjectures for S-arithmetic groups II’. Unpublished preprint.
  34. Ji L. ‘Buildings and their applications in geometry and topology’. Asian J. Math. (2006) 10:11–80.
  35. Jost J. Nonpositive curvature: Geometric and analytic aspects. (Birkhäuser, Klosterberg, 1997).
  36. Landvogt E. ‘Some functorial properties of the Bruhat–Tits building’. J. Reine Angew. Math. (2000) 518:213–241.
  37. Lück W. ‘Survey on classifying spaces of familes of subgroups’. Infinite groups: Geometric, combinatorial and dynamical aspects. 269–322. Progress in Mathematics 248 (Birkhäuser, Klosterberg, 2005).
  38. Lück W., Reich H. ‘The Baum-Connes and the Farrell-Jones conjectures in K- and L-theory’. Handbook of K-theory. 703–842. vol. 1, 2 (Springer, Berlin, 2005).
  39. Margulis G. Discrete subgroups of semisimple Lie groups. (Springer, Berlin, 1991) x+388.
  40. Matsnev D. ‘The Baum-Connes conjecture and group actions on affine buildings’. (2005) Thesis, Pennsylvania State University, University Park.
  41. Platonov V., Rapinchuk A. Algebraic groups and number theory. Pure and Applied Mathematics 139 (Academic Press, New York, 1994).
  42. Prasad G. ‘Semi-simple groups and arithmetic subgroups’. 821–832. Proceedings of the International Congress of Mathematicians (Mathematical Society of Japan, Tokyo, 1991).
  43. Ranicki A. ‘Algebraic L-theory and topological manifolds’. Cambridge Tracts in Mathematics. 102. (Cambridge University Press, Cambridge, 1992).
  44. Ranicki A. Lower K- and L-theory. London Mathematical Society Lecture Note Series 178. (Cambridge University Press, Cambridge, 1992).
  45. Rosenberg J. ‘Review of The Novikov conjecture: Geometry and algebra by M. Kreck and W. Lück’. Bull. Amer. Math. Soc. (2006) 43:599–604.[CrossRef]
  46. Rosenthal D. ‘Splitting with continuous control in algebraic K-theory’. K-Theory (2004) 32:139–166.
  47. Rosenthal D. ‘Continuous control and the algebraic L-theory assembly map’. Forum Math. (2006) 18:193–209.[CrossRef]
  48. Serre J. P. ‘Arithmetic groups’. Homological group theory. (Cambridge University Press, Cambridge, 1979).
  49. Serre J. P. ‘Cohomologie des groupes discrets’. Prospects in Mathematics. 77–169. Annals of Mathematics Studies 70 (Princeton University Press, Princeton, 1972).
  50. Siebenmann L. ‘Infinite simple homotopy types’. Indag. Math. (1970) 32:479–495.
  51. Yu G. ‘The Novikov conjecture for groups with finite asymptotic dimension’. Ann. Math. (1998) 147:325–355.[CrossRef]

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This Article
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