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Tits type alternative for groups acting on toric affine varieties

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arxiv 2003.00037 v3 pith:3VYYMHA4 submitted 2020-02-28 math.AG math.GR

I. Arzhantsev , M. Zaidenberg This is my paper · ORCID
classification math.AGmath.GR
keywords actingaffinealgebraicalternativecontainsfreegroupldots
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abstract

Given a toric affine algebraic variety $X$ and a collection of one-parameter unipotent subgroups $U_1,\ldots,U_s$ of $\mathop{\rm Aut}(X)$ which are normalized by the torus acting on $X$, we show that the group $G$ generated by $U_1,\ldots,U_s$ 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.

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