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BRANCH GROUPS, ORBIT GROWTH, AND SUBGROUP GROWTH TYPES FOR PRO-
$p$ GROUPS
Published online by Cambridge University Press: 26 May 2020
Abstract
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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 
$n^{\log n}$.
MSC classification
- Type
- Algebra
- Information
- Creative Commons
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.- Copyright
- © The Author(s) 2020
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