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Commensurability growth of branch groups. (English) Zbl 1514.20011

Summary: Fixing a subgroup \(\Gamma\) in a group \(G\), the commensurability growth function assigns to each \(n\) the cardinality of the set of subgroups \(\Delta\) of \(G\) with \([\Gamma: \Gamma \cap \Delta][\Delta : \Gamma \cap \Delta] = n\). For pairs \(\Gamma \leq A\), where \(A\) is the automorphism group of a \(p\)-regular rooted tree and \(\Gamma\) is finitely generated, we show that this function can take on finite, countable, or uncountable cardinals. For almost all known branch groups \(\Gamma \) (the first Grigorchuk group, the twisted twin Grigorchuk group, Pervova groups, Gupta-Sidki groups, etc.) acting on \(p\)-regular rooted trees, this function is precisely \(\aleph_0\) for any \(n = p^k\).

MSC:

20B07 General theory for infinite permutation groups
20E26 Residual properties and generalizations; residually finite groups
20K10 Torsion groups, primary groups and generalized primary groups
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References:

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