Free \(\mathbb{Q}\)-groups are residually torsion-free nilpotent. (Les \(Q\)-groupes libres sont résiduellement nilpotents sans torsion.) (English. French summary) Zbl 07904674
A group \(G\) is called a \(\mathbb{Q}\)-group if, for any \(n\in \mathbb{N}\) and \(g \in G\) there, exists exactly one \(h \in G\) satisfying \(h^{n}=g\). These groups were introduced by G. Baumslag in [Acta Math. 104, 217–303 (1960; Zbl 0178.34901)], he observed that \(\mathbb{Q}\)-groups may be viewed as universal algebras, and as such they constitute a variety. In particular the free algebras (in that variety) are called free \(\mathbb{Q}\)-groups.
The main result in the article under review is Theorem 1.1: A free \(\mathbb{Q}\)-group is residually torsion-free nilpotent. This affirmatively answers a question posed by G. Baumslag in [Commun. Pure Appl. Math. 18, 25–30 (1965; Zbl 0136.01204)].
The key point in the proof of Theorem 1.1 is to show that any finitely generated subgroup of a free \(\mathbb{Q}\)-group can be embedded into a finitely generated free pro-\(p\) group for some prime \(p\). Theorem 1.1 is actually an application of the more technical Theorem 5.1, the proof of which uses the theory of mod-\(p\; L^{2}\)-Betti numbers.
Another interesting result of this paper is given by Theorem 1.4: The \(\mathbb{Q}\)-completion of a limit group is residually torsion-free nilpotent.
The main result in the article under review is Theorem 1.1: A free \(\mathbb{Q}\)-group is residually torsion-free nilpotent. This affirmatively answers a question posed by G. Baumslag in [Commun. Pure Appl. Math. 18, 25–30 (1965; Zbl 0136.01204)].
The key point in the proof of Theorem 1.1 is to show that any finitely generated subgroup of a free \(\mathbb{Q}\)-group can be embedded into a finitely generated free pro-\(p\) group for some prime \(p\). Theorem 1.1 is actually an application of the more technical Theorem 5.1, the proof of which uses the theory of mod-\(p\; L^{2}\)-Betti numbers.
Another interesting result of this paper is given by Theorem 1.4: The \(\mathbb{Q}\)-completion of a limit group is residually torsion-free nilpotent.
Reviewer: Egle Bettio (Venezia)
MSC:
| 20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |
| 20E18 | Limits, profinite groups |
| 20E26 | Residual properties and generalizations; residually finite groups |
Keywords:
free \(\mathbb{Q}\)-group; free pro-\(p\) group; mod-\(p\) \(L^2\)-Betti numbers; residually finite group; residually nilpotent group; \(\mathbb{Q}\)-completion; universal division ring of fractionsReferences:
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