A note on normal generation and generation of groups

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 $n_1,\dots ,n_k\in \{1,2,\dots \}\cup \{\mathrm{\infty }\}$, then $${\beta }_1^{(2)}(G)\le k-1-\sum _{i=1}^k\frac{1}{n_i},$$ where ${\beta }_1^{(2)}(G)$ denotes the first ${\mathrm{\ell }}^2$-Betti number of $G$. We also show that any $k$-generated group with ${\beta }_1^{(2)}(G)\ge k-1-\epsilon$ must have girth greater than or equal $1\mathrm{/}\epsilon$.