Tits type alternative for groups acting on toric affine varieties

Abstract

Given a toric affine algebraic variety X and a collection of one-parameter unipotent subgroups U 1 , . . . , U s of Aut(X) which are normalized by the torus acting on X, we show that the group G generated by U 1 , . . . , U s verifies the Tits alternative, and, moreover, either is a unipotent algebraic group, or contains a nonabelian free subgroup. We deduce that if G is m-transitive for any positive integer m, then G contains a nonabelian free subgroup, and so, is of exponential growth.

Some finiteness results on triangular automorphisms

“…, G s of connected triangular algebraic subgroups of the group Aut(A n ) generates a connected solvable algebraic subgroup of Aut(A n ) . This note originates from an attempt to prove [2,Proposition 3.6]. Let G a be the additive group of the ground filed or, equivalently, a one-dimensional unipotent algebraic group.…”

“…Let G a be the additive group of the ground filed or, equivalently, a one-dimensional unipotent algebraic group. In order to show that a finite collection of triangular G a -subgroups generates a unipotent algebraic subgroup we used in [2] specific arguments including the Baker-Campbell-Hausdorff formula and the multivariate Zassenhaus formula. There is a desire to prove this fact by more direct methods.…”