Finite group extensions and the Atiyah conjecture

Linnell, Peter¹ ; Schick, Thomas²

¹ Department of Mathematics, Virginia Tech, Blacksburg, Virginia 24061-0123
² Mathematisches Institut, Georg-August-Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany

Journal of the American Mathematical Society, Tome 20 (2007) no. 4, pp. 1003-1051

Résumé

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 http://www.grouptheory.info. 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.

Détail

DOI : 10.1090/S0894-0347-07-00561-9

Bibliographie

[1] Adem, Alejandro, Milgram, R. James Cohomology of finite groups 1994

[2] Atiyah, M. F. Elliptic operators, discrete groups and von Neumann algebras 1976 43 72

[3] Baumslag, G., Dyer, E., Heller, A. The topology of discrete groups J. Pure Appl. Algebra 1980 1 47

[4] Botto Mura, Roberta, Rhemtulla, Akbar Orderable groups 1977

[5] Bourbaki, Nicolas Commutative algebra. Chapters 1–7 1989

[6] Brown, Kenneth S. Cohomology of groups 1982

[7] Cohn, P. M. Free rings and their relations 1985

[8] Dicks, Warren, Schick, Thomas The spectral measure of certain elements of the complex group ring of a wreath product Geom. Dedicata 2002 121 137

[9] Dodziuk, Jozef de Rham-Hodge theory for 𝐿²-cohomology of infinite coverings Topology 1977 157 165

[10] Dodziuk, Jã³Zef, Linnell, Peter, Mathai, Varghese, Schick, Thomas, Yates, Stuart Approximating 𝐿²-invariants and the Atiyah conjecture Comm. Pure Appl. Math. 2003 839 873

[11] Dummit, David S., Foote, Richard M. Abstract algebra 1991

[12] Kohno, Toshitake Série de Poincaré-Koszul associée aux groupes de tresses pures Invent. Math. 1985 57 75

[13] Falk, Michael, Randell, Richard Pure braid groups and products of free groups 1988 217 228

[14] Fenn, R., Greene, M. T., Rolfsen, D., Rourke, C., Wiest, B. Ordering the braid groups Pacific J. Math. 1999 49 74

[15] Grigorchuk, Rostislav I., Linnell, Peter, Schick, Thomas, ŻUk, Andrzej On a question of Atiyah C. R. Acad. Sci. Paris Sér. I Math. 2000 663 668

[16] Gromov, M. Kähler hyperbolicity and 𝐿₂-Hodge theory J. Differential Geom. 1991 263 292

[17] Gruenberg, K. W. Residual properties of infinite soluble groups Proc. London Math. Soc. (3) 1957 29 62

[18] Hall, P. Some sufficient conditions for a group to be nilpotent Illinois J. Math. 1958 787 801

[19] Hempel, John Residual finiteness for 3-manifolds 1987 379 396

[20] Hughes, Ian Division rings of fractions for group rings Comm. Pure Appl. Math. 1970 181 188

[21] Hughes, Ian Division rings of fractions for group rings. II Comm. Pure Appl. Math. 1972 127 131

[22] Humphries, Stephen P. Torsion-free quotients of braid groups Internat. J. Algebra Comput. 2001 363 373

[23] Jackowski, Stefan A fixed-point theorem for 𝑝-group actions Proc. Amer. Math. Soc. 1988 205 208

[24] Jaco, William Lectures on three-manifold topology 1980

[25] Jacobson, Nathan Basic algebra. II 1989

[26] Kochman, S. O. Bordism, stable homotopy and Adams spectral sequences 1996

[27] Kropholler, P. H., Linnell, P. A., Moody, J. A. Applications of a new 𝐾-theoretic theorem to soluble group rings Proc. Amer. Math. Soc. 1988 675 684

[28] Labute, John P. Algèbres de Lie et pro-𝑝-groupes définis par une seule relation Invent. Math. 1967 142 158

[29] Labute, John P. On the descending central series of groups with a single defining relation J. Algebra 1970 16 23

[30] Labute, John P. The Lie algebra associated to the lower central series of a link group and Murasugi’s conjecture Proc. Amer. Math. Soc. 1990 951 956

[31] Linnell, Peter A. Division rings and group von Neumann algebras Forum Math. 1993 561 576

[32] Linnell, Peter A. Analytic versions of the zero divisor conjecture 1998 209 248

[33] Lã¼Ck, Wolfgang Transformation groups and algebraic 𝐾-theory 1989

[34] Lã¼Ck, Wolfgang 𝐿²-invariants of regular coverings of compact manifolds and CW-complexes 2002 735 817

[35] Lã¼Ck, Wolfgang, Schick, Thomas, Thielmann, Thomas Torsion and fibrations J. Reine Angew. Math. 1998 1 33

[36] Lyndon, R. C. Two notes on nilpotent groups Proc. Amer. Math. Soc. 1952 579 583

[37] Papakyriakopoulos, C. D. On Dehn’s lemma and the asphericity of knots Ann. of Math. (2) 1957 1 26

[38] Passman, Donald S. Infinite crossed products 1989

[39] Rolfsen, Dale, Zhu, Jun Braids, orderings and zero divisors J. Knot Theory Ramifications 1998 837 841

[40] Schick, Thomas Integrality of 𝐿²-Betti numbers Math. Ann. 2000 727 750

[41] Schick, Thomas Integrality of 𝐿²-Betti numbers Math. Ann. 2000 727 750

[42] Serre, Jean-Pierre Cohomologie galoisienne 1994

[43] Serre, Jean-Pierre Galois cohomology 1997

[44] Stallings, John Homology and central series of groups J. Algebra 1965 170 181

[45] Switzer, Robert M. Algebraic topology—homotopy and homology 1975

[46] Wilson, John S. Profinite groups 1998