Combinatorics of the tame automorphism group

Lamy, Stéphane

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 28 (2019) no. 1, pp. 145-207.

Nous étudions le groupe $\mathrm{Tame}\left({𝔸}^3\right)$ des automorphismes modérés de l’espace affine de dimension 3, sur un corps de caractéristique nulle. Nous retrouvons, de manière unifiée, des résultats de Kuroda, Shestakov, Umirbaev et Wright, concernant la théorie des réductions et les relations dans $\mathrm{Tame}\left({𝔸}^3\right)$. La nouveauté dans notre approche réside dans la mise en avant d’un complexe simplicial de dimension 2 simplement connexe sur lequel $\mathrm{Tame}\left({𝔸}^3\right)$ agit par isométries.

We study the group $\mathrm{Tame}\left({𝔸}^3\right)$ of tame automorphisms of the 3-dimensional affine space, over a field of characteristic zero. We recover, in a unified way, previous results of Kuroda, Shestakov, Umirbaev and Wright, about the theory of reduction and the relations in $\mathrm{Tame}\left({𝔸}^3\right)$. The novelty in our presentation is the emphasis on a simply connected 2-dimensional simplicial complex on which $\mathrm{Tame}\left({𝔸}^3\right)$ acts by isometries.

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