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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
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[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 \(P\)^10,1,8, 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
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