##
**An analogue of the Levi decomposition of the automorphism groups of certain nilpotent pro-\(\ell\) groups.**
*(English)*
Zbl 0705.20035

Let G be a finitely generated nilpotent pro-\(\ell\) group, \(\{G_ k|\) \(k\in {\mathbb{N}}\}\) its descending central series with \(G_ k/G_{k+1}\) being a free \({\mathbb{Z}}_{\ell}\)-module of finite rank k. Denote by m the least integer with \(G_ m=(1)\). The author proves Theorem: for \(\ell \geq m\) and the group \(\Omega\) of bi-continuous automorphisms of G the short exact sequence \(1\to Ker \sigma \to \Omega^{\sigma}\to Aut(G/G_ 2)\to 1\) (with \(\sigma\) being the canonical homomorphism) splits. He notices also that there exists an automorphism \(\sigma_{\alpha}\in \Omega\) such that \(x_ i^{\sigma_{\alpha}}=x_ i^{\alpha}\) (1\(\leq i\leq r)\) for a given generating set \(\{x_ 1,...,x_ r\}\) of G and \(\alpha \in {\mathbb{Z}}^*_{\ell}\) satisfying \(\alpha^ j\neq 1\) (1\(\leq j\leq m-2)\). The author shows that the centralizer \(C_{\Omega}(\sigma_{\sigma})\) is independent of \(\alpha\) and this subgroup \(\Pi =C_{\Omega}(\sigma_{\alpha})\) is such that \(\Pi\cap Ker \sigma =(1)\) and \(Im(\sigma |_{\Pi})=Aut(G/G_ 2)\). The author notices also that \(\Omega\) can be viewed as a linear \(\ell\)-adic Lie group and \(C(\sigma_{\alpha})\) as its Levi subgroup. Two remarks are added: (1) for \(m>\ell\) the above theorem isn’t true in general, and (2) there exist hopes to give some application of the theorem to Galois representations.

Reviewer: U.Kaljulaid

### MSC:

20F28 | Automorphism groups of groups |

20E18 | Limits, profinite groups |

20F14 | Derived series, central series, and generalizations for groups |

22E20 | General properties and structure of other Lie groups |

### Keywords:

finitely generated nilpotent pro-\(\ell \) group; descending central series; bi-continuous automorphisms; generating set; linear \(\ell \)-adic Lie group; Levi subgroup
Full Text:
DOI

### References:

[1] | Asada, M; Kaneko, M, On the automorphism group of some pro-l fundamental groups, (), 137-159 |

[2] | Bourbaki, N, Groupes et algebres de Lie, (1972), Hermann Paris, Chap. 2 et 3 · Zbl 0244.22007 |

[3] | Deligne, P, A letter to S. Bloch, (February, 1984) |

[4] | Ihara, Y, Profinite braid groups, Galois representations and complex multiplications, Ann. of math., 123, 43-106, (1986) · Zbl 0595.12003 |

[5] | Ihara, Y, On Galois representations arising from towers of coverings of \(P\)^{1}0,1,8, Invent. math., 86, 427-459, (1987) |

[6] | Ihara, Y, Some problems on three point ramifications and associated large Galois representations, (), 173-188 |

[7] | Kaneko, M, (), [Japanese] |

[8] | Kaneko, M, On conjugacy classes of the pro-l braid group of degree 2, (), 274-277 · Zbl 0618.20022 |

[9] | Kohno, T; Oda, T, The lower central series of the pure braid group of an algebraic curve, (), 201-219 |

[10] | Lazard, M, Sur LES groupes nilpotents et LES anneaux de Lie, Ann. sci. ecole norm. sup., 71, 101-190, (1954), (3) · Zbl 0055.25103 |

[11] | Lazard, M, Groupes analytiques p-adiques, Inst. hautes etudes sci. publ. math., 26, (1965) · Zbl 0139.02302 |

[12] | Oda, T, Two propositions on pro-l braid groups, (1985), preprint |

[13] | Oda, T, Note on meta-abelian quotients of pro-l free groups, (1985), preprint |

[14] | Weil, A, Sur LES groupes à pn éléments, Rev. sci., 77, 321-322, (1939), (Collected papers, Vol. I, pp. 241-243) · JFM 65.1126.01 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.