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 $U_1,\dots ,U_s$ of $\mathrm{Aut}(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\dots ,U_s$ verifies the following alternative of Tits' type: either $G$ is a unipotent algebraic group, or it contains a non-abelian free subgroup. We deduce that if $G$ is $2$-transitive on a $G$-orbit in $X$, then $G$ contains a non-abelian free subgroup, and so, is of exponential growth.

24 pages. The main result strengthened, the proof of Proposition 4.8 written in more detail; some references added; the referee remarks taken into account; the title changed

References in corpus (1)

• On multi-variable Zassenhaus formula

Cited by in corpus (5)

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• Automorphisms of algebraic varieties and infinite transitivity
• Some finiteness results on triangular automorphisms
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• On finite-dimensional homogeneous Lie algebras of derivations of polynomial rings