Abstract
The normal rank of a group is the minimal number of elements whose normal closure coincides with the group. We study the relation between the normal rank of a group and its first \(\ell ^2\)-Betti number and conjecture the inequality \(\beta _1^{(2)}(G) \le \mathrm{nrk}(G)-1\) for torsion free groups. The conjecture is proved for limits of left-orderable amenable groups. On the other hand, for every \(n\ge 2\) and every \(\varepsilon >0\), we give an example of a simple group \(Q\) (with torsion) such that \(\beta _1^{(2)}(Q) \ge n-1-\varepsilon \). These groups also provide examples of simple groups of rank exactly \(n\) for every \(n\ge 2\); existence of such examples for \(n> 3\) was unknown until now.
Similar content being viewed by others
References
Arzhantseva, G., Minasyan, A., Osin, D.V.: The SQ-universality and residual properties of relatively hyperbolic groups. J. Algebra 315(1), 165–177 (2007)
Atiyah, M.F.: Elliptic operators, discrete groups and von Neumann algebras. In: Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), pp. 4372. Asterisque, No. 32–33, Soc. Math. France, Paris, 1976.
Baumslag, G., Myasnikov, A.G., Shpilrain, V.: Open problems in combinatorial group theory. Second edition. Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), 138, Contemp. Math., 296, Am. Math. Soc., Providence, RI, 2002
Baumslag, G., Roseblade, J.E.: Subgroups of direct products of free groups. J. Lond. Math. Soc. 30, 44–52 (1984)
Bekka, B.M., Valette, A.: Group cohomology, harmonic functions and the first \(L^2\)-Betti number. Potential Anal. 6(4), 313–326 (1997)
Bergman, G.M.: Right orderable groups that are not locally indicable. Pac. J. Math. 147(2), 243–248 (1991)
Berrick, J., Hillman, J.: The Whitehead conjecture and \(\ell ^2\)-Betti numbers, Guido’s Book of Conjectures, Monographies de L’Enseignement Mathématique, L’Enseignement Mathématique. 35–37 (2008)
Bridson, M.R., Tweedale, M.: Deficiency and abelianized deficiency of some virtually free groups. Math. Proc. Camb. Phil. Soc. 143, 257–264 (2007)
Cheeger, J., Gromov, M.: \(L_2\)-cohomology and group cohomology. Topology 25(2), 189–215 (1986)
Duchamp, G., Krob, D.: The lower central series of the free partially commutative group. Semigr. Forum 45(3), 385–394 (1992)
Elek, G.: On the analytic zero divisor conjecture of Linnell. Bull. Lond. Math. Soc. 35(2), 236–238 (2003)
Gaboriau, D.: Coût des relations d’équivalence et des groupes. Invent. Math. 139, 41–98 (2000)
Gersten, S.: Conservative groups, indicability, and a conjecture of Howie. J. Pure Appl. Algebra 29(1), 59–74 (1983)
Gromov, M.: Hyperbolic groups. In: Gersten, S.M. (ed.) Essays in group theory, MSRI Series, vol. 8, p. 263. Springer, Berlin (1987)
Gromov, M.: Asymptotic invariants of infinite groups. Geometric group theory, vol. 2. In: Niblo, Roller, (ed.) Proc. Symp. Sussex Univ., Brighton, July 1419, 1991, Lond. Math. Soc. Lecture Notes, vol. 182. Cambridge Univ. Press, Cambridge (1993)
Groves, D., Manning, J.: Dehn filling in relatively hyperbolic groups. Isr. J. Math. 168, 317–429 (2008)
Guba, V.: A finitely generated simple group with free 2-generated subgroups. Sib. Math. J. 27(5), 670–684 (1986)
Lackenby, M.: Expanders, rank and graphs of groups. Isr. J. Math. 146, 357–370 (2005)
Lennox, J.C., Wiegold, J.: Generators and killers for direct and free products. Arch. Math. 34(4), 296–300 (1980)
W. Lück, \(L^2\)-invariants: theory and applications to geometry and \(K\)-theory. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. In: A series of modern surveys in mathematics, vol. 44. Springer, Berlin, (2002).
Lück, W.: Approximating \(L^2\)-invariants by their finite-dimensional analogues. Geom. Funct. Anal. 4(4), 455–481 (1994)
Lück, W., Osin, D.: Approximating the first \(L^2\)-Betti number of residually finite groups. J. Top. Anal. 3(2), 153–160 (2011)
Morris, D.: Amenable groups that act on the line. Algebr. Geom. Topol. 6, 2509–2518 (2006)
Obraztsov, V.N.: Growth sequences of 2-generator simple groups. Proc. R. Soc. Edinb. 123(5), 839–855 (1993)
Olshanskii, A.Yu.: Geometry of defining relations in groups. Kluwer Academic Pub-lisher, Dordrecht (1991)
Olshanskii, AYu.: On residualing homomorphisms and \(G\)-subgroups of hyperbolic groups. Int. J. Algebr. Comput. 3(4), 365–409 (1993)
Osin, D.: Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems. Memoirs Am. Math. Soc. 179 (2006), no. 843
Osin, D.: Elementary subgroups of relatively hyperbolic groups and bounded generation. Int. J. Algebr. Comput. 16(1), 99–118 (2006)
Osin, D.: Peripheral fillings of relatively hyperbolic groups. Invent. Math. 167(2), 295–326 (2007)
Osin, D.: Small cancellations over relatively hyperbolic groups and embedding theorems. Ann. Math. 172(1), 1–39 (2010)
Peterson, J., Thom, A.: Group cocycles and the ring of affiliated operators. Invent. Math. 185(3), 561–592 (2011)
Pichot, M.: Semi-continuity of the first \(l^2\)-Betti number on the space of finitely generated groups. Comment. Math. Helv. 81(3), 643–652 (2006)
Thom, A.: Sofic groups and diophantine approximation. Comm. Pure Appl. Math. LXI, 1155–1171 (2008)
Author information
Authors and Affiliations
Corresponding author
Additional information
The research of D. Osin was supported by the NSF Grant DMS-1006345 and by the RFBR Grant 11-01-00945.
Rights and permissions
About this article
Cite this article
Osin, D., Thom, A. Normal generation and \(\ell ^2\)-Betti numbers of groups. Math. Ann. 355, 1331–1347 (2013). https://doi.org/10.1007/s00208-012-0828-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-012-0828-7