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 $n_1,\dots,n_k \in \{1,2,\dots \} \cup \{\infty \}$, then $β_1^{(2)}(G) \leq k-1-\sum_{i=1}^{k} \frac1{n_i},$ where $β_1^{(2)}(G)$ denotes the first $\ell^2$-Betti number of $G$. We also show that any $k$-generated group with $β_1^{(2)}(G) \geq k-1-\varepsilon$ must have girth greater than or equal $1/\varepsilon$.

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