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Some reflections on proving groups residually torsion-free nilpotent. I. (English) Zbl 1250.20021
A group \(G\) is called ‘residually torsion-free nilpotent’ if the intersection of its normal subgroups with torsion-free nilpotent quotients is trivial. In 1935, Wilhelm Magnus introduced associative algebras together with topology into the mix making it possible to prove that free groups are residually torsion-free nilpotent.
The objective of the paper is to give some alternative proofs of Magnus’ theorem. This approach gives rise to a new family of one-relator groups with this property. The family includes, among others, the fundamental groups of orientable surfaces. The author hopes that this approach will lead to new insights into these well-known groups, and a proof that free \(\mathbb Q\)-groups have the same property.

MSC:
20E26 Residual properties and generalizations; residually finite groups
20E05 Free nonabelian groups
20F14 Derived series, central series, and generalizations for groups
20F18 Nilpotent groups
20F05 Generators, relations, and presentations of groups
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Full Text: Euclid
References:
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