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arXiv · 2003.00037

Tits type alternative for groups acting on toric affine varieties

I. Arzhantsev·M. Zaidenberg
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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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I. Arzhantsev, M. Zaidenberg. 2020-02-28. Tits type alternative for groups acting on toric affine varieties. https://doi.org/10.1093/imrn%2Frnaa342

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