Lower central series and split extensions
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arXiv
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| Formato: | Preprint |
| Publicado: |
2021
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| _version_ | 1866908890233307136 |
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| author | Darné, Jacques Suciu, Alexander I. |
| author_facet | Darné, Jacques Suciu, Alexander I. |
| contents | Following Lazard, we study the $N$-series of a group $G$ and their associated graded Lie algebras. The main examples we consider are the lower central series (LCS), Stallings' rational and mod-$q$ versions, and Zassenhaus' mod-$p$ version of the LCS. We treat them as part of a general construction of the $\mathcal P$-LCS, for a property $\mathcal P$ of filtrations. We describe these $N$-series and the associated Lie algebras in the case when $G$ splits as a semi-direct product, in terms of the relevant data for the factors and the monodromy action. This allows us to generalize the well-known theorem of Falk-Randell regarding the LCS of split extensions to other versions of the LCS. In particular, we generalize the mod-$q$ version of Bellingeri-Gervais to any integer $q$, and we prove analogous results for the rational LCS and Zassenhaus' mod-$p$ LCS.
We then use the same tools to study residual properties of semi-direct products, and how they interact with residual properties of the factors. We also give a new proof of a classical theorem of Gruenberg. Finally, we apply our results to surface braid groups, which naturally split as semi-direct products, allowing us to recover and generalize known results about the residual nilpotency of groups of pure braids on surfaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2105_14129 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Lower central series and split extensions Darné, Jacques Suciu, Alexander I. Group Theory Geometric Topology 17B70, 20F14, 20F36, 20F40, 20J05, 57M07 Following Lazard, we study the $N$-series of a group $G$ and their associated graded Lie algebras. The main examples we consider are the lower central series (LCS), Stallings' rational and mod-$q$ versions, and Zassenhaus' mod-$p$ version of the LCS. We treat them as part of a general construction of the $\mathcal P$-LCS, for a property $\mathcal P$ of filtrations. We describe these $N$-series and the associated Lie algebras in the case when $G$ splits as a semi-direct product, in terms of the relevant data for the factors and the monodromy action. This allows us to generalize the well-known theorem of Falk-Randell regarding the LCS of split extensions to other versions of the LCS. In particular, we generalize the mod-$q$ version of Bellingeri-Gervais to any integer $q$, and we prove analogous results for the rational LCS and Zassenhaus' mod-$p$ LCS. We then use the same tools to study residual properties of semi-direct products, and how they interact with residual properties of the factors. We also give a new proof of a classical theorem of Gruenberg. Finally, we apply our results to surface braid groups, which naturally split as semi-direct products, allowing us to recover and generalize known results about the residual nilpotency of groups of pure braids on surfaces. |
| title | Lower central series and split extensions |
| topic | Group Theory Geometric Topology 17B70, 20F14, 20F36, 20F40, 20J05, 57M07 |
| url | https://arxiv.org/abs/2105.14129 |