A note on normal generation and generation of groups

Abstract: 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 $$\beta_1^{(2)}(G) \leq k-1-\sum_{i=1}^{k} \frac1{n_i},$$ 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 $1/\varepsilon$.