Normal generation and ell ^2-Betti numbers of groups

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 $nge 2$ 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 $nge 2$ ; existence of such examples for $n> 3$ was unknown until now.