The group Aut($\mathbb{A}^{n}$) of polynomial automorphisms of the a ne space is an interesting huge group, and a slightly simpler group is its subgroup Aut($\mathbb{A}^{n}$) of tame automorphisms. Natural questions about these groups include:– does they admit normal subgroups beside the obvious subgroup of automorphisms with Jacobian 1?– do they satisfy a Tits alternative?– what are the possible dynamical degrees of their elements? The method to investigate these questions is via some actions on some metric spaces, namely the coset complex and the valuation complex, that we plan to introduce in detail. The lectures will focus on the following three cases: the group Aut($\mathbb{A}^{2}$) = Tame($\mathbb{A}^{2}$) following Chapter 7 of my book in preparation, then the group Tame($\mathbb{A}^{3}$) (Lamy, LamyPrzytycki, Blanc-Van Santen), and finally the group Tame(Q($\mathbb{A}^{4}$)) of tame automorphisms of $\mathbb{A}^{4}$ preserving a nondegenerate quadratic form (Bisi-Furter-Lamy, Martin, Dang).[-]

The group of tame automorphisms. The group Aut($\mathbb{A}^{n}$) of polynomial automorphisms of the a ne space is an interesting huge group, and a slightly simpler group is its subgroup Aut($\mathbb{A}^{n}$) of tame automorphisms. Natural questions about these groups include:– does they admit normal subgroups beside the obvious subgroup of automorphisms with Jacobian 1?– do they satisfy a Tits alternative?– what are the possible dynamical degrees of their elements? The method to investigate these questions is via some actions on some metric spaces, namely the coset complex and the valuation complex, that we plan to introduce in detail. The lectures will focus on the following three cases: the group Aut($\mathbb{A}^{2}$) = Tame($\mathbb{A}^{2}$) following Chapter 7 of my book in preparation, then the group Tame($\mathbb{A}^{3}$) (Lamy, LamyPrzytycki, Blanc-Van Santen), and finally the group Tame(Q($\mathbb{A}^{4}$)) of tame automorphisms of $\mathbb{A}^{4}$ preserving a nondegenerate quadratic form (Bisi-Furter-Lamy, Martin, Dang).[-]

The group of tame automorphisms. The group Aut($\mathbb{A}^{n}$) of polynomial automorphisms of the a ne space is an interesting huge group, and a slightly simpler group is its subgroup Aut($\mathbb{A}^{n}$) of tame automorphisms. Natural questions about these groups include:– does they admit normal subgroups beside the obvious subgroup of automorphisms with Jacobian 1?– do they satisfy a Tits alternative?– what are the possible dynamical degrees of their elements? The method to investigate these questions is via some actions on some metric spaces, namely the coset complex and the valuation complex, that we plan to introduce in detail. The lectures will focus on the following three cases: the group Aut($\mathbb{A}^{2}$) = Tame($\mathbb{A}^{2}$) following Chapter 7 of my book in preparation, then the group Tame($\mathbb{A}^{3}$) (Lamy, LamyPrzytycki, Blanc-Van Santen), and finally the group Tame(Q($\mathbb{A}^{4}$)) of tame automorphisms of $\mathbb{A}^{4}$ preserving a nondegenerate quadratic form (Bisi-Furter-Lamy, Martin, Dang).[-]