On the maximality of the triangular subgroup

Furter, Jean-Philippe; Poloni, Pierre-Marie

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 $\mathrm{Aut}\left({𝔸}_{ℂ}^n\right)$ pour tout $n$. Il forme ainsi un sous-groupe de Borel du ind-groupe $\mathrm{Aut}\left({𝔸}_{ℂ}^n\right)$. 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 $\mathrm{Aut}\left({𝔸}_{ℂ}^2\right)$. 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 $\mathrm{Aut}\left({𝔸}_{ℂ}^2\right)$ tout entier ?

We prove that the subgroup of triangular automorphisms of the complex affine $n$-space is maximal among all solvable subgroups of $\mathrm{Aut}\left({𝔸}_{ℂ}^n\right)$ for every $n$. In particular, it is a Borel subgroup of $\mathrm{Aut}\left({𝔸}_{ℂ}^n\right)$, 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 $\mathrm{Aut}\left({𝔸}_{ℂ}^2\right)$. 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 $\mathrm{Aut}\left({𝔸}_{ℂ}^2\right)$.

Classification : 14R10, 20G99
Keywords: Polynomial automorphisms, triangular automorphisms, ind-groups
Mot clés : Automorphismes polynomiaux, automorphismes triangulaires, ind-groupes

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