no. 1
On the maximality of the triangular subgroup
[Sur la maximalité du sous-groupe triangulaire]
Furter, Jean-Philippe1 ; Poloni, Pierre-Marie2
1 Université de La Rochelle Laboratoire MIA avenue Michel Crépeau 17000 La Rochelle (France)
2 Universität Bern Mathematisches Institut Sidlerstrasse 5 CH-3012 Bern (Switzerland)
Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 393-421.

Nous montrons que le sous-groupe des automorphismes triangulaires est un sous-groupe résoluble maximal de Aut(𝔸 ℂ n ) pour tout n. Il forme ainsi un sous-groupe de Borel du ind-groupe Aut(𝔸 ℂ n ). En dimension deux, nous montrons que le sous-groupe triangulaire est un sous-groupe fermé maximal mais qu’il n’est néanmoins pas maximal parmi tous les sous-groupes de Aut(𝔸 ℂ 2 ). Un automorphisme f de 𝔸 ℂ 2 étant donné, nous étudions la question suivante : le sous-groupe engendré par f et par les automorphismes triangulaires est-il égal au groupe Aut(𝔸 ℂ 2 ) tout entier ?

We prove that the subgroup of triangular automorphisms of the complex affine n-space is maximal among all solvable subgroups of Aut(𝔸 ℂ n ) for every n. In particular, it is a Borel subgroup of Aut(𝔸 ℂ n ), when the latter is viewed as an ind-group. In dimension two, we prove that the triangular subgroup is a maximal closed subgroup and that nevertheless, it is not maximal among all subgroups of Aut(𝔸 ℂ 2 ). Given an automorphism f of 𝔸 ℂ 2 , we study the question whether the group generated by f and the triangular subgroup is equal to the whole group Aut(𝔸 ℂ 2 ).

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Révisé le :
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DOI : 10.5802/aif.3165
Classification : 14R10, 20G99
Keywords: Polynomial automorphisms, triangular automorphisms, ind-groups
Mot clés : Automorphismes polynomiaux, automorphismes triangulaires, ind-groupes

Furter, Jean-Philippe 1 ; Poloni, Pierre-Marie 2

1 Université de La Rochelle Laboratoire MIA avenue Michel Crépeau 17000 La Rochelle (France)
2 Universität Bern Mathematisches Institut Sidlerstrasse 5 CH-3012 Bern (Switzerland)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Furter, Jean-Philippe; Poloni, Pierre-Marie. On the maximality of the triangular subgroup. Annales de l'Institut Fourier, Tome 68 (2018) no. 1, pp. 393-421. doi : 10.5802/aif.3165. https://aif.centre-mersenne.org/articles/10.5802/aif.3165/

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