In their book Subgroup Growth, Lubotzky and Segal asked: What are the possible types of subgroup growth of the pro- $p$ group? In this paper, we construct certain extensions of the Grigorchuk group and the Gupta–Sidki groups, which have all possible types of subgroup growth between $n^{(\log n)^{2}}$ and $e^{n}$ . Thus, we give an almost complete answer to Lubotzky and Segal’s question. In addition, we show that a class of pro- $p$ branch groups, including the Grigorchuk group and the Gupta–Sidki groups, all have subgroup growth type $n^{\log n}$ .

中文翻译：

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专业组的分支组、轨道增长和子组增长类型

在他们的书中亚群增长, Lubotzky 和 ​​Segal 问道：亲子群体增长的可能类型是什么？ $p$ 团体？在本文中，我们构建了 Grigorchuk 群和 Gupta-Sidki 群的某些扩展，它们在 $n^{(\log n)^{2}}$ 和 $e^{n}$ . 因此，我们对 Lubotzky 和 ​​Segal 的问题给出了几乎完整的答案。此外，我们证明了一类亲 $p$ 分支群，包括 Grigorchuk 群和 Gupta-Sidki 群，都具有亚群增长类型 $n^{\log n}$ .