A note on normal generation and generation of groups

summary:In this note we study sets of normal generators of finitely presented residually $p$-finite groups. We show that if an infinite, finitely presented, residually $p$-finite group $G$ is normally generated by $g_1,\dots ,g_k$ with order $n_1,\dots ,n_k \in \{1,2,\dots \} \cup \{\infty \}$, then $$\beta _1^{(2)}(G) \leq k-1-\sum _{i=1}^{k} \frac 1{n_i}\,,$$ where $\beta _1^{(2)}(G)$ denotes the first $\ell ^2$-Betti number of $G$. We also show that any $k$-generated group with $\beta _1^{(2)}(G) \geq k-1-\varepsilon $ must have girth greater than or equal $1/\varepsilon $