On the maximality of the triangular subgroup
Annales de l'Institut Fourier, Volume 68 (2018) no. 1, p. 393-421
We prove that the subgroup of triangular automorphisms of the complex affine n-space is maximal among all solvable subgroups of Aut(𝔸 n ) for every n. In particular, it is a Borel subgroup of Aut(𝔸 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 2 ). Given an automorphism f of 𝔸 2 , we study the question whether the group generated by f and the triangular subgroup is equal to the whole group Aut(𝔸 2 ).
Nous montrons que le sous-groupe des automorphismes triangulaires est un sous-groupe résoluble maximal de Aut(𝔸 n ) pour tout n. Il forme ainsi un sous-groupe de Borel du ind-groupe Aut(𝔸 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 2 ). Un automorphisme f de 𝔸 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 2 ) tout entier ?
Received : 2016-07-11
Revised : 2017-04-07
Accepted : 2017-09-14
Published online : 2018-04-18
DOI : https://doi.org/10.5802/aif.3165
Classification:  14R10,  20G99
Keywords: Polynomial automorphisms, triangular automorphisms, ind-groups
@article{AIF_2018__68_1_393_0,
     author = {Furter, Jean-Philippe and Poloni, Pierre-Marie},
     title = {On the maximality of the triangular subgroup},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {68},
     number = {1},
     year = {2018},
     pages = {393-421},
     doi = {10.5802/aif.3165},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2018__68_1_393_0}
}
On the maximality of the triangular subgroup. Annales de l'Institut Fourier, Volume 68 (2018) no. 1, pp. 393-421. doi : 10.5802/aif.3165. https://aif.centre-mersenne.org/item/AIF_2018__68_1_393_0/

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