On the maximality of the triangular subgroup

Abstract

We prove that the subgroup of triangular automorphisms of the complex affine n-space is maximal among all solvable subgroups of Aut(A n C ) for every n. In particular, it is a Borel subgroup of Aut(A n C ), when the latter is viewed as an indgroup. 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 C ). Given an automorphism f of A 2 C , we study the question whether the group generated by f and the triangular subgroup is equal to the whole group Aut(A 2 C ).