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Normal generation and \(\ell ^2\)-Betti numbers of groups

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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.

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Authors and Affiliations

  1. Department of Mathematics, Vanderbilt University, Nashville, TN, USA

    Denis Osin

  2. Universität Leipzig, Leipzig, Germany

    Andreas Thom

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  1. Denis Osin
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  2. Andreas Thom
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Correspondence to Andreas Thom.

Additional information

The research of D. Osin was supported by the NSF Grant DMS-1006345 and by the RFBR Grant 11-01-00945.

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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

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