Galois-theoretic features for 1-smooth pro-p groups [article]

Claudio Quadrelli
2021 arXiv   pre-print
Let p be a prime. A pro-p group G is said to be 1-smooth if it can be endowed with a continuous representation θ G→GL_1(ℤ_p) such that every open subgroup H of G, together with the restriction θ|_H, satisfies a formal version of Hilbert 90. We prove that every 1-smooth pro-p group contains a unique maximal closed abelian normal subgroup, in analogy with a result by Engler and Koenigsmann on maximal pro-p Galois groups of fields, and that if a 1-smooth pro-p group is solvable, then it is locally
more » ... uniformly powerful, in analogy with a result by Ware on maximal pro-p Galois groups of fields. Finally we ask whether 1-smooth pro-p groups satisfy a "Tits' alternative".
arXiv:2004.12605v5 fatcat:hoacka7v5je6ngek4zgc7tjk7u