关于群的正规生成与生成的注记
A note on normal generation and generation of groups
摘要：
本文研究了剩余$p$ -有限群的正规生成元集.证明了若一个无限的、可表示的、剩余的$p$ -有限群$G$是由阶为$n_1,\dots,n_k \in \{1,2,\dots \} \cup \{\infty \}$的$g_1,\dots,g_k$正规生成的，则有$$β_1^{(2)}(G) \leq k-1-\sum_{i=1}^{k} \frac1{n_i},$$，其中$β_1^{(2)}(G)$表示$G$的第一个$\ell^2$ - Betti数.我们还证明了任何具有$β_1^{(2)}(G) \geq k-1-\varepsilon$的$k$生成群的围长必须大于或等于$1/\varepsilon$ .
英文摘要：
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 $1/\varepsilon$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
