Abstract
LetX be a finite connectedCW-complex. Suppose that its fundamental group π is residually finite, i.e. there is a nested sequence ... ⊂ Г m + 1 ⊂ Г m ⊂ ... ⊂ π of in π normal subgroups of finite index whose intersection is trivial. Then we show that thep-thL 2-Betti number ofX is the limit of the sequenceb p(Xm)/[π:Г m ] whereb p(Xm) is the (ordinary)p-th Betti number of the finite covering ofX associated with Г m .
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
[A]M. Atiyah, Elliptic operators, discrete groups and von Neumann algebras, Asterisque 32 (1976), 43–72.
[B]H. Bass, Euler characteristics and characters of discrete groups, Invent. Math. 35 (1976), 155–196.
[Ba]G. Baumslag, Automorphism groups of residually finite groups, J. LMS 38 (1963), 117–118.
[CMat]A.L. Carey, V. Mathai,L 2-torsion invariants, preprint (1991).
[ChG]J. Cheeger, M. Gromov,L 2-cohomology and group cohomology, Topology 25 (1986), 189–215.
[Co]M.M. Cohen, Combinatorial group theory: a topological approach, LMS student texts 14 (1989).
[D]J. Dodziuk, DeRham-Hodge theory forL 2-cohomology of infinite coverings, Topology 16 (1977), 157–165.
[DoX]H. Donnelly, F. Xavier, On the differential form spectrum of negatively curved Riemannian manifolds, Amer. J. Math. 106 (1984), 169–185.
[E]A. Efremov, Combinatorial and Analytic Novikov Shubin invariant, preprint (1991).
[FJ]F.T. Farrell, L.E. Jones, Isomorphism conjectures in algebraicK-theory, J. of AMS 6 (1993), 249–298.
[FuK]B. Fuglede, R.V. Kadison, Determinant theory in finite factors, Ann. of Math. 55 (1952), 520–530.
[G1M. Gromov, Volume and bounded cohomology, Publ. Math. IHES 56 (1982), 5–100.
[G2]M. Gromov, Kähler hyperbolicity andL 2-Hodge theory, J. of Diff. Geom. 33 (1991), 263–292.
[G3]M. Gromov, Asymptotic invariants of infinite groups, preprint, IHES/M/92 /8, to appear in “Geometric Groups Theory Volume 2”, Proc. of the Symp. in Sussex 1991, edited by G.A. Niblo and M.A. Roller, Lecture Notes Series 182, Cambridge University Press (1992).
[GS]M. Gromov, M.A. Shubin, Von Neumann spectra near zero, Geometric And Functional Analysis 1 (1991), 375–404.
[Gr]K.W. Grünberg, Residual properties of infinite soluble groups, Proc LMS 7 (1957), 29–62.
[H]J. Hempel, Residual finiteness for 3-manifolds, in “Combinatoral Group Theory and Topology”, (S.M. Gersten, J.R. Stallings, eds.), 379–396, Annals of Mathematics Studies 111, Princeton University Press, Princeton (1987).
[Hi]G. Higman, A finitely generated infinite simple group, J. LMS 26 (1951), 61–64.
[KR]R.V. Kadison, J.R. Ringrose, Fundamentals of the theory of operator algebras, volume II: Advanced theory, Pure and Applied Mathematics, Academic Press (1986).
[Ka]D. Kazhdan, On arithmetic varieties, in “Lie Groups and Their Representations”, Halsted (1975), 151–216.
[L]J. Lott, Heat kernels on covering spaces and topological invariants, J. of Diff. Geom. 35 (1992), 471–510.
[LLü]J. Lott, W. Lück,L 2-topological invariants of 3-manifolds, preprint, MPI Bonn (1992).
[Lü1]W. Lück,L 2-torsion and 3-manifolds, preprint, to appear in the Proceedings for the Conference “Low-dimensional topology” (K. Johannson, ed), Knoxville 1992, International Press Co. Cambridge, MA 02238 (1992).
[Lü2]W. Lück,L 2-Betti numbers of mapping tori and groups, preprint (1992), Topology 33 (1994), 203–214.
[LüRo]W. Lück, M. Rothenberg, Reidemeister torsion and theK-theory of von Neumann algebras,K-theory 5 (1991), 213–264.
[M]W. Magnus, Residually finite groups, Bull. AMS 75 (1969), 305–315.
[Ma]A. Mal'cev, On isomorphic matrix representations of finite groups, Mat. Sb. 8 (1940), 405–422.
[Mat]V. Mathai,L 2-analytic torsion, preprint (1990).
[Mo]A.W. Mostowski, On the decidability of some problems in special classes of groups, Fund. Math. 59 (1966), 123–135.
[N]H. Neumann, Varieties of groups, Ergebnisse der Mathematik und ihre Grenzgebiete 37, Springer (1967).
[S]R.G. Swan, Induced representations and projective modules, Ann. of Math. 71 (1960), 552–578.
[W]B.A.F. Wehrfritz, Infinite linear groups, Ergebnisse der Mathematik und ihrer Grenzgebiete 76, Springer (1973).
[Y]S.K. Yeung, Betti numbers on a tower of coverings, Duke Math. J. 73 (1994), 201–226.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Lück, W. ApproximatingL 2-invariants by their finite-dimensional analogues. Geometric and Functional Analysis 4, 455–481 (1994). https://doi.org/10.1007/BF01896404
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01896404