The Novikov conjecture for linear groups
Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 243-268.

Let K be a field. We show that every countable subgroup of GL (n,K) is uniformly embeddable in a Hilbert space. This implies that Novikov’s higher signature conjecture holds for these groups. We also show that every countable subgroup of GL (2,K) admits a proper, affine isometric action on a Hilbert space. This implies that the Baum-Connes conjecture holds for these groups. Finally, we show that every subgroup of GL (n,K) is exact, in the sense of C * -algebra theory.

@article{PMIHES_2005__101__243_0,
     author = {Guentner, Erik and Higson, Nigel and Weinberger, Shmuel},
     title = {The {Novikov} conjecture for linear groups},
     journal = {Publications Math\'ematiques de l'IH\'ES},
     pages = {243--268},
     publisher = {Springer},
     volume = {101},
     year = {2005},
     doi = {10.1007/s10240-005-0030-5},
     mrnumber = {2217050},
     zbl = {1073.19003},
     language = {en},
     url = {http://www.numdam.org/articles/10.1007/s10240-005-0030-5/}
}
TY  - JOUR
AU  - Guentner, Erik
AU  - Higson, Nigel
AU  - Weinberger, Shmuel
TI  - The Novikov conjecture for linear groups
JO  - Publications Mathématiques de l'IHÉS
PY  - 2005
SP  - 243
EP  - 268
VL  - 101
PB  - Springer
UR  - http://www.numdam.org/articles/10.1007/s10240-005-0030-5/
DO  - 10.1007/s10240-005-0030-5
LA  - en
ID  - PMIHES_2005__101__243_0
ER  - 
%0 Journal Article
%A Guentner, Erik
%A Higson, Nigel
%A Weinberger, Shmuel
%T The Novikov conjecture for linear groups
%J Publications Mathématiques de l'IHÉS
%D 2005
%P 243-268
%V 101
%I Springer
%U http://www.numdam.org/articles/10.1007/s10240-005-0030-5/
%R 10.1007/s10240-005-0030-5
%G en
%F PMIHES_2005__101__243_0
Guentner, Erik; Higson, Nigel; Weinberger, Shmuel. The Novikov conjecture for linear groups. Publications Mathématiques de l'IHÉS, Tome 101 (2005), pp. 243-268. doi : 10.1007/s10240-005-0030-5. http://www.numdam.org/articles/10.1007/s10240-005-0030-5/

1. R. Alperin, P. Shalen, Linear groups of finite cohomological dimension. Invent. Math., 66 (1982), 89-98. | MR | Zbl

2. C. Anantharaman-Delaroche, Amenability and exactness for dynamical systems and their C*-algebras. Trans. Amer. Math. Soc., 354 (2002), 4153-4178 (electronic). | MR | Zbl

3. M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral asymmetry and Riemannian geometry, I. Math. Proc. Cambridge Philos. Soc., 77 (1975), 43-69. | MR | Zbl

4. M. F. Atiyah, V. K. Patodi, I. M. Singer, Spectral asymmetry and Riemannian geometry, II. Math. Proc. Cambridge Philos. Soc., 78 (1975), 405-432. | MR | Zbl

5. N. Bourbaki, Commutative algebra. Berlin: Springer 1989.

6. P. Baum, A. Connes, N. Higson, Classifying space for proper actions and K-theory of group C*-algebras, in C*-algebras: 1943-1993 (San Antonio, TX, 1993), vol. 167 of Contemp. Math., pp. 240-291. Providence, RI: Am. Math. Soc. 1994. | MR | Zbl

7. M. E. B. Bekka, P.-A. Cherix, A. Valette, Proper affine isometric actions of amenable groups, in Novikov conjectures, index theorems and rigidity, vol. 2 (Oberwolfach, 1993), vol. 227 of London Math. Soc. Lecture Note Ser., pp. 1-4. Cambridge: Cambridge Univ. Press 1995. | MR | Zbl

8. K. S. Brown, Buildings. New York: Springer 1989. | MR | Zbl

9. J. W. S. Cassels, Local fields, vol. 3 of London Mathematical Society Student Texts. Cambridge: Cambridge University Press 1986. | MR | Zbl

10. P.-A. Cherix, M. Cowling, P. Jolissaint, P. Julg, A. Valette, Groups with the Haagerup property, vol. 197 of Progress in Mathematics. Basel: Birkhäuser 2001. Gromov's a-T-menability. | Zbl

11. M. Dadarlat, E. Guentner, Constructions preserving Hilbert space uniform embeddability of discrete groups. Trans. Amer. Math. Soc., 355 (2003), 3235-3275. | MR | Zbl

12. P. De La Harpe, A. Valette, La propriété (T) de Kazhdan pour les groupes localement compacts (avec un appendice de Marc Burger). Astérisque, 175 (1989). With an appendix by M. Burger. | Numdam | MR | Zbl

13. P. B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, vol. 11 of Mathematics Lecture Series. Publish or Perish, Inc. 1984. | MR | Zbl

14. M. Gromov, Asymptotic invariants of infinite groups, in Geometric group theory, vol. 2 (Sussex, 1991), vol. 182 of London Math. Soc. Lecture Note Ser., pp. 1-295. Cambrige: Cambridge Univ. Press 1993. | MR | Zbl

15. E. Guentner, J. Kaminker, Exactness and the Novikov conjecture and addendum. Topology, 41 (2002), 411-419. | MR | Zbl

16. U. Haagerup, An example of a nonnuclear C*-algebra, which has the metric approximation property. Invent. Math., 50 (1978/79), 279-293. | MR | Zbl

17. S. Helgason, Differential Geometry, Lie Groups, and Symmetric Spaces, vol. 80 of Pure and Applied Mathematics. New York: Academic Press 1978. | MR | Zbl

18. N. Higson, Bivariant K-theory and the Novikov conjecture. Geom. Funct. Anal., 10 (2000), 563-581. | MR | Zbl

19. N. Higson, G. Kasparov, Operator K-theory for groups which act properly and isometrically on Hilbert space. Electron. Res. Announc. Amer. Math. Soc., 3 (1997), 131-142 (electronic). | MR | Zbl

20. N. Higson, G. Kasparov, E-theory and KK-theory for groups which act properly and isometrically on Hilbert space. Invent. Math., 144 (2001), 23-74. | MR | Zbl

21. G. G. Kasparov, Equivariant KK-theory and the Novikov conjecture. Invent. Math., 91 (1988), 147-201. | MR | Zbl

22. G. G. Kasparov, G. Skandalis, Groups acting on buildings, operator K-theory, and Novikov's conjecture. K-Theory, 4 (1991), 303-337. | Zbl

23. E. Kirchberg, S. Wassermann, Permanence properties of C*-exact groups. Doc. Math., 4 (1999), 513-558 (electronic). | MR | Zbl

24. S. Lang, Algebra, 2nd edn. Menlo Park, CA: Addison-Wesley 1984. | MR | Zbl

25. W. Lück, R. Stamm, Computations of K- and L-theory of cocompact planar groups. K-Theory, 21 (2000), 249-292. | MR | Zbl

26. I. Madsen, R. J. Milgram, The classifying spaces of surgery and cobordism of topological manifolds. Princeton Univ. Press 1970. | Zbl

27. N. Ozawa, Amenable actions and exactness for discrete groups. C. R. Acad. Sci. Paris, Sér. I, Math., 330 (2000), 691-695. | MR | Zbl

28. M. Pimsner, D. Voiculescu, Exact sequences for K-groups and Ext-groups of certain cross-product C*-algebras. J. Oper. Theory, 4 (1980), 93-118. | MR | Zbl

29. J. P. Serre, Trees. Translated from the French by J. Stillwell. Berlin: Springer 1980. | MR | Zbl

30. G. Skandalis, J. L. Tu, G. Yu, The coarse Baum-Connes conjecture and groupoids. Topology, 41 (2002), 807-834. | MR | Zbl

31. S. Wassermann, C*- exact groups, in C*-algebras (Münster, 1999), pp. 243-249. Berlin: Springer 2000. | MR | Zbl

32. S. Weinberger, Homotopy invariance of η-invariants. Proc. Natl. Acad. Sci., 85 (1988), 5362-5363. | Zbl

33. S. Weinberger, Rationality of ρ-invariants. Math. Z., 223 (1996), 197-246. Appendix to “Jumps of the eta-invariant”, by M. Farber and J. Levine. | Zbl

34. G. Yu, The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space. Invent. Math., 139 (2000), 201-240. | MR | Zbl

35. R. J. Zimmer, Kazhdan groups acting on compact manifolds. Invent. Math., 75 (1984), 425-436. | MR | Zbl

Cité par Sources :