Abstract
The goal of this chapter is to determine normal forms for elements in finitely generated free groups and free nilpotent groups. This is done by using a collection process which we discuss in Sect. 3.1. A study of weighted commutators and basic commutators in a group relative to a given generating set also appears in this section. The highlight of Sect. 3.1 is a fundamental result stating that if G is any group generated by a set X and γ i G denotes the ith lower central subgroup, then each quotient γ n G/γ n+1 G is generated, modulo γ n+1 G, by a sequence of basic commutators on X of weight n. Section 3.2 is devoted to the so-called collection formula. This formula expresses a positive power of a product of elements x 1, …, x r of a group as a product of positive powers of basic commutators in x 1, …, x r . The collection process developed in Sect. 3.1 plays a key role here. In Sect. 3.3, we investigate basic commutators in finitely generated free groups and free nilpotent groups. We prove a major result which states that a finitely generated free nilpotent group, freely generated by a set X, has a “basis” consisting of basic commutators in X. The techniques used in this section involve groupoids, Lie rings, and the Magnus embedding. We end the chapter with Sect. 3.4, which is devoted to the rather technical proof of the collection formula obtained in Sect. 3.2.
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Clement, A.E., Majewicz, S., Zyman, M. (2017). The Collection Process and Basic Commutators. In: The Theory of Nilpotent Groups. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-66213-8_3
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DOI: https://doi.org/10.1007/978-3-319-66213-8_3
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