Abstract
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 $n^{\log n}$ .
Highlights
Introduction and resultsFor the rest of this paper, p is a fixed prime and log n = logp n
Given a function f : N → R, we say that G has subgroup growth of type f (n) if there exist a, b > 0 such that sn(G) f (n)a for all n and f (n)b sn(G) for infinitely many n
We show that all functions in the range n(log n)2 to en occur as the subgroup growth type of a pro- p group
Summary
What types of subgroup growth of pro- p groups exist between nlog n and n(log n)2 ? We are not sure whether the answer to the second part is positive, that is, we will not be surprised if there exists a log-concave function f such that there are no pro- p groups of strict growth type f. We comment that previously known examples of pro- p groups with subgroup growth type nlog n are of different nature, that is, either linear and analytic groups or the Nottingham group and its index subgroups Many of these examples are just infinite. Ershov and Jaikin-Zapirain in [2] constructed new hereditarily just infinite pro- p groups These groups have subgroup growth type at least n(log n)2− , where is any positive number (private communication).
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