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 $n\ge 2$ 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.