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Finite group extensions and the Atiyah conjecture

Authors: Peter Linnell and Thomas Schick
Journal: J. Amer. Math. Soc. 20 (2007), 1003-1051
MSC (2000): Primary 55N25, 16S34, 57M25
Published electronically: March 14, 2007
MathSciNet review: 2328714
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Abstract: The Atiyah conjecture for a discrete group $ G$ states that the $ L^2$-Betti numbers of a finite CW-complex with fundamental group $ G$ are integers if $ G$ is torsion-free, and in general that they are rational numbers with denominators determined by the finite subgroups of $ G$.

Here we establish conditions under which the Atiyah conjecture for a torsion-free group $ G$ implies the Atiyah conjecture for every finite extension of $ G$. The most important requirement is that $ H^*(G,\mathbb{Z}/p)$ is isomorphic to the cohomology of the $ p$-adic completion of $ G$ for every prime number $ p$. An additional assumption is necessary e.g. that the quotients of the lower central series or of the derived series are torsion-free.

We prove that these conditions are fulfilled for a certain class of groups, which contains in particular Artin's pure braid groups (and more generally fundamental groups of fiber-type arrangements), free groups, fundamental groups of orientable compact surfaces, certain knot and link groups, certain primitive one-relator groups, and products of these. Therefore every finite, in fact every elementary amenable extension of these groups satisfies the Atiyah conjecture, provided the group does.

As a consequence, if such an extension $ H$ is torsion-free, then the group ring $ \mathbb{C}H$ contains no non-trivial zero divisors, i.e. $ H$ fulfills the zero-divisor conjecture.

In the course of the proof we prove that if these extensions are torsion-free, then they have plenty of non-trivial torsion-free quotients which are virtually nilpotent. All of this applies in particular to Artin's full braid group, therefore answering question B6 on

Our methods also apply to the Baum-Connes conjecture. This is discussed by Thomas Schick in his preprint ``Finite group extensions and the Baum-Connes conjecture'', where for example the Baum-Connes conjecture is proved for the full braid groups.

References [Enhancements On Off] (What's this?)

  • 1. Alejandro Adem and R. James Milgram.
    Cohomology of finite groups.
    Springer-Verlag, Berlin, 1994. MR 1317096 (96f:20082)
  • 2. M. F. Atiyah.
    Elliptic operators, discrete groups and von Neumann algebras.
    In Colloque ``Analyse et Topologie'' en l'Honneur de Henri Cartan (Orsay, 1974), pages 43-72. Astérisque, No. 32-33. Soc. Math. France, Paris, 1976. MR 0420729 (54:8741)
  • 3. G. Baumslag, E. Dyer, and A. Heller.
    The topology of discrete groups.
    J. Pure Appl. Algebra, 16(1):1-47, 1980. MR 549702 (81i:55012)
  • 4. Roberta Botto Mura and Akbar Rhemtulla.
    Orderable groups.
    Marcel Dekker Inc., New York, 1977.
    Lecture Notes in Pure and Applied Mathematics, Vol. 27. MR 0491396 (58:10652)
  • 5. Nicolas Bourbaki.
    Commutative algebra. Chapters 1-7.
    Springer-Verlag, Berlin, 1989.
    Translated from the French, Reprint of the 1972 edition. MR 0979760 (90a:13001)
  • 6. Clemens Bratzler.
    $ L^2$-Betti Zahlen und Faserungen.
    Diplomarbeit, Universität Mainz, 1997.
  • 7. Kenneth S. Brown.
    Cohomology of groups.
    Springer-Verlag, New York, 1982. MR 672956 (83k:20002)
  • 8. P. M. Cohn.
    Free rings and their relations.
    Academic Press Inc. [Harcourt Brace Jovanovich Publishers], London, second edition, 1985. MR 800091 (87e:16006)
  • 9. Warren Dicks and Thomas Schick.
    The spectral measure of certain elements of the complex group ring of a wreath product.
    Geom. Dedicata, 93:121-137, 2002. MR 1934693 (2003i:20005)
  • 10. Jozef Dodziuk.
    De Rham-Hodge theory for $ {L}\sp{2}$-cohomology of infinite coverings.
    Topology, 16(2):157-165, 1977. MR 0445560 (56:3898)
  • 11. Jozef Dódziuk, Peter Linnell, Varghese Mathai, Thomas Schick, and Stuart Yates.
    Approximating $ {L}^2$-invariants, and the Atiyah conjecture.
    Comm. Pure Appl. Math., 56(7):839-873, 2003. MR 1990479 (2004g:58040)
  • 12. David S. Dummit and Richard M. Foote.
    Abstract algebra.
    Prentice Hall Inc., Englewood Cliffs, NJ, 2nd. edition, 1994. MR 1138725 (92k:00007)
  • 13. Michael Falk and Richard Randell.
    The lower central series of a fiber-type arrangement.
    Invent. Math., 82(1):77-88, 1985. MR 808110 (87c:32015b)
  • 14. Michael Falk and Richard Randell.
    Pure braid groups and products of free groups.
    In Braids (Santa Cruz, CA, 1986), pages 217-228. Amer. Math. Soc., Providence, RI, 1988. MR 975081 (90d:20070)
  • 15. R. Fenn, M. T. Greene, D. Rolfsen, C. Rourke, and B. Wiest.
    Ordering the braid groups.
    Pacific J. Math., 191(1):49-74, 1999. MR 1725462 (2000j:20064)
  • 16. Rostislav I. Grigorchuk, Peter Linnell, Thomas Schick, and Andrzej Zuk.
    On a question of Atiyah.
    C. R. Acad. Sci. Paris Sér. I Math., 331(9):663-668, 2000. MR 1797748 (2001m:57050)
  • 17. Michael Gromov.
    Kähler hyperbolicity and $ {L}\sb 2$-Hodge theory.
    J. Differential Geom., 33(1):263-292, 1991. MR 1085144 (92a:58133)
  • 18. K. W. Gruenberg.
    Residual properties of infinite soluble groups.
    Proc. London Math. Soc. (3), 7:29-62, 1957. MR 0087652 (19:386a)
  • 19. P. Hall.
    Some sufficient conditions for a group to be nilpotent.
    Illinois J. Math., 2:787-801, 1958. MR 0105441 (21:4183)
  • 20. John Hempel.
    Residual finiteness for $ 3$-manifolds.
    In Combinatorial group theory and topology (Alta, Utah, 1984), pages 379-396. Princeton Univ. Press, Princeton, NJ, 1987. MR 895623 (89b:57002)
  • 21. K. A. Hirsch.
    On infinite soluble groups. II.
    Proc. London Math. Soc. (2), 44:336-344, 1938.
  • 22. Ian Hughes.
    Division rings of fractions for group rings.
    Comm. Pure Appl. Math., 23:181-188, 1970. MR 0263934 (41:8533)
  • 23. Ian Hughes.
    Division rings of fractions for group rings. II.
    Comm. Pure Appl. Math., 25:127-131, 1972. MR 0292956 (45:2037)
  • 24. Stephen P. Humphries.
    Torsion-free quotients of braid groups.
    Internat. J. Algebra Comput., 11(3):363-373, 2001. MR 1847185 (2002f:20054)
  • 25. Stefan Jackowski.
    A fixed-point theorem for $ p$-group actions.
    Proc. Amer. Math. Soc., 102(1):205-208, 1988. MR 915745 (89a:57052)
  • 26. William Jaco.
    Lectures on three-manifold topology.
    American Mathematical Society, Providence, R.I., 1980. MR 565450 (81k:57009)
  • 27. Nathan Jacobson.
    Basic algebra. II.
    W. H. Freeman and Company, New York, second edition, 1989. MR 1009787 (90m:00007)
  • 28. S. O. Kochman.
    Bordism, stable homotopy and Adams spectral sequences.
    American Mathematical Society, Providence, RI, 1996. MR 1407034 (97i:55017)
  • 29. P. H. Kropholler, P. A. Linnell, and J. A. Moody.
    Applications of a new $ {K}$-theoretic theorem to soluble group rings.
    Proc. Amer. Math. Soc., 104(3):675-684, 1988. MR 964842 (89j:16016)
  • 30. Inga Kümpel, Peter Linnell, and Thomas Schick.
    Galois cohomology of completed link groups.
    in preparation.
  • 31. John P. Labute.
    Algèbres de Lie et pro-$ p$-groupes définis par une seule relation.
    Invent. Math., 4:142-158, 1967. MR 0218495 (36:1581)
  • 32. John P. Labute.
    On the descending central series of groups with a single defining relation.
    J. Algebra, 14:16-23, 1970. MR 0251111 (40:4342)
  • 33. John P. Labute.
    The Lie algebra associated to the lower central series of a link group and Murasugi's conjecture.
    Proc. Amer. Math. Soc., 109(4):951-956, 1990. MR 1013973 (90k:20065)
  • 34. V. Lin.
    Braids, permutations, polynomials I.
    Preprint, Max Planck Institut für Mathematik, Bonn, 112 pages, 1996.
  • 35. Peter A. Linnell.
    Division rings and group von Neumann algebras.
    Forum Math., 5(6):561-576, 1993. MR 1242889 (94h:20009)
  • 36. Peter A. Linnell.
    Analytic versions of the zero divisor conjecture.
    In Geometry and cohomology in group theory (Durham, 1994), pages 209-248. Cambridge Univ. Press, Cambridge, 1998. MR 1709960 (2000g:20016)
  • 37. Wolfgang Lück.
    Transformation groups and algebraic $ {K}$-theory, volume 1408 of Lecture Notes in Mathematics.
    Springer-Verlag, Berlin, 1989.
    Mathematica Gottingensis. MR 1027600 (91g:57036)
  • 38. Wolfgang Lück.
    $ L\sp 2$-invariants of regular coverings of compact manifolds and CW-complexes.
    In Handbook of geometric topology, pages 735-817. North-Holland, Amsterdam, 2002. MR 1886681 (2003a:58049)
  • 39. Wolfgang Lück, Thomas Schick, and Thomas Thielmann.
    Torsion and fibrations.
    J. Reine Angew. Math., 498:1-33, 1998. MR 1629917 (99e:58203)
  • 40. R. C. Lyndon.
    Two notes on nilpotent groups.
    Proc. Amer. Math. Soc., 3:579-583, 1952. MR 0049889 (14:242a)
  • 41. Wilhelm Magnus.
    Über beziehungen zwischen höheren Kommutatoren.
    J. Reine Angew. Math., 177:105-115, 1937.
  • 42. C. D. Papakyriakopoulos.
    On Dehn's lemma and the asphericity of knots.
    Ann. of Math. (2), 66:1-26, 1957. MR 0090053 (19:761a)
  • 43. Donald S. Passman.
    Infinite crossed products.
    Academic Press Inc., Boston, MA, 1989. MR 979094 (90g:16002)
  • 44. Holger Reich.
    Group von Neumann algebras and related algebras.
    Ph.D. thesis, Universität Göttingen, 1999.
  • 45. Dale Rolfsen and Jun Zhu.
    Braids, orderings and zero divisors.
    J. Knot Theory Ramifications, 7(6):837-841, 1998. MR 1643939 (99g:20072)
  • 46. Thomas Schick.
    Finite group extensions and the Baum-Connes conjecture.
    preprint, available via http://arXiv/math.KT/0209165.
  • 47. Thomas Schick.
    Integrality of $ {L}\sp 2$-Betti numbers.
    Math. Ann., 317(4):727-750, 2000. MR 1777117 (2002k:55009a)
  • 48. Thomas Schick.
    Erratum: ``Integrality of $ L\sp 2$-Betti numbers''.
    Math. Ann., 322(2):421-422, 2002. MR 1894160 (2002k:55009b)
  • 49. Jean-Pierre Serre.
    Cohomologie Galoisienne.
    Springer-Verlag, Berlin, fifth edition, 1994. MR 1324577 (96b:12010)
  • 50. Jean-Pierre Serre.
    Galois cohomology.
    Springer-Verlag, Berlin, 1997.
    Translated from the French by Patrick Ion and revised by the author. MR 1466966 (98g:12007)
  • 51. John Stallings.
    Homology and central series of groups.
    J. Algebra, 2:170-181, 1965. MR 0175956 (31:232)
  • 52. Robert M. Switzer.
    Algebraic topology--homotopy and homology.
    Springer-Verlag, New York, 1975.
    Die Grundlehren der mathematischen Wissenschaften, Band 212. MR 0385836 (52:6695)
  • 53. John S. Wilson.
    Profinite groups.
    The Clarendon Press Oxford University Press, New York, 1998. MR 1691054 (2000j:20048)

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Additional Information

Peter Linnell
Affiliation: Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123

Thomas Schick
Affiliation: Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany

Received by editor(s): May 31, 2005
Published electronically: March 14, 2007
Additional Notes: The first author was partially supported by SFB 478, Münster
Research of the second author was funded by DAAD (German Academic Exchange Agency)
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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