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Almost all finitely generated subgroups of the Nottingham group are free. (English) Zbl 1073.20022

The Nottingham group over the prime field \(\mathbb{F}_p\) is the group \(N(p)\) of formal power series \(\{t(1+a_1t+a_2t^2+\cdots )\mid a_i\in\mathbb{F}_p\}\) under substitution. It is a pro-\(p\) group and satisfies the relevant property that every finite \(p\)-group can be embedded in \(N(p)\). It follows that the free group of rank 2 can also be embedded in \(N(p)\) and this observation led A. Shalev [in M. du Sautoy (ed.), et al., New horizons in pro-\(p\) groups. Boston, MA: Birkhäuser. Prog. Math. 184, 1-54 (2000; Zbl 0981.20020)] to ask whether it is true that almost all the pairs of elements of \(N(p)\) generate an abstract free group.
In this nice paper the author shows that the answer is positive, giving a short and elementary proof of the following fact: two random elements of the Nottingham group \(N(p)\) generate a free group with probability 1. As a consequence he also proves that \(N(p)\) contains a dense free subgroup generated by two elements.

MSC:

20E18 Limits, profinite groups
13F25 Formal power series rings
20E07 Subgroup theorems; subgroup growth
20P05 Probabilistic methods in group theory
20F05 Generators, relations, and presentations of groups

Citations:

Zbl 0981.20020
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