Some extreme properties of the affine group as an automophism group of the affine space (Q2759822)

For a field \(k\), the affine Cremona group \(\text{GA}_n\) is the group of all \(k\)-algebra automorphisms of the polynomial ring \(k[x_1,\dots,x_n]\). Let \(\text{AGL}_n\) denote the subgroup of \(\text{GA}_n\) consisting of all linear automorphisms, the ordinary affine group of the affine space \(A^n\). It is known that \(\text{GA}_n\) carries the structure of an infinite-dimensional algebraic group and its corresponding Lie algebra \(\text{gl}_n\) is an irreducible transitive graded algebra of finite growth. By using this correspondence, the author of this paper describes the structure of the closed subgroups between \(\text{GL}_n\) and \(\text{GA}_n\). In particular, the author proves that \(\text{AGL}_n\) is a maximal closed subgroup of \(\text{GA}_n\). In addition, it is shown that for any natural number \(m\), each subgroup of \(\text{GA}_n\) containing \(\text{AGL}_n\) acts \(m\)-transitively on \(A^n\).NEWLINENEWLINEFor the entire collection see [Zbl 0970.00014].