Filtrations of free groups arising from the lower central series
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Michael Chapman
und
Ido Efrat
Abstract
We make a systematic study of filtrations of a free group F defined as products of powers of the lower central series of F. Under some assumptions on the exponents, we characterize these filtrations in terms of the group algebra, the Magnus algebra of non-commutative power series, and linear representations by upper-triangular unipotent matrices. These characterizations generalize classical results of Grün, Magnus, Witt, and Zassenhaus from the 1930s, as well as later results on the lower p-central filtration and the p-Zassenhaus filtrations. We derive alternative recursive definitions of such filtrations, extending results of Lazard. Finally, we relate these filtrations to Massey products in group cohomology.
Funding source: Israel Science Foundation
Award Identifier / Grant number: 23/09
Funding source: Israel Science Foundation
Award Identifier / Grant number: 152/13
Funding statement: This work was supported by the Israel Science Foundation (grants No. 23/09 and 152/13). It is partially based on the M.Sc. thesis of the first author done under the supervision of the second author in Ben-Gurion University of the Negev.
We thank Ilya Tyomkin for valuable remarks which shortened some of our original arguments. We also thank the anonymous referee for his/her very helpful comments, which we have used to improve the presentation at several points.
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© 2016 by De Gruyter
Artikel in diesem Heft
- Frontmatter
- Sliceable groups and towers of fields
- Pro-Hall R-groups and groups discriminated by the free pro-p group
- Filtrations of free groups arising from the lower central series
- Prosolvability criteria and properties of the prosolvable radical via Sylow sequences
- Non-commutative lattice problems
- Conjugacy distinguished subgroups
- Diophantine questions in the class of finitely generated nilpotent groups
- Uncountably many non-commensurable finitely presented pro-p groups
- Singer torus in irreducible representations of GL(n,q)
- Virtually free pro-p products
Artikel in diesem Heft
- Frontmatter
- Sliceable groups and towers of fields
- Pro-Hall R-groups and groups discriminated by the free pro-p group
- Filtrations of free groups arising from the lower central series
- Prosolvability criteria and properties of the prosolvable radical via Sylow sequences
- Non-commutative lattice problems
- Conjugacy distinguished subgroups
- Diophantine questions in the class of finitely generated nilpotent groups
- Uncountably many non-commensurable finitely presented pro-p groups
- Singer torus in irreducible representations of GL(n,q)
- Virtually free pro-p products
