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arXiv · 2111.06659

Tits-type alternative for certain groups acting on algebraic surfaces

Ivan Arzhantsev·Mikhail Zaidenberg
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Abstract

A theorem of Cantat and Urech says that an analog of the classical Tits alternative holds for the group of birational automorphisms of a compact complex Kaehler surface. We established in our previous paper the following Tits-type alternative: if X is a toric affine variety and G is a subgroup of Aut(X) generated by a finite set of unipotent subgroups normalized by the acting torus then either G contains a nonabelian free subgroup or G is a unipotent affine algebraic group. In the present paper we extend the latter result to any group G of automorphisms of a complex affine surface generated by a finite collection of unipotent algebraic subgroups. It occurs that either G contains a nonabelian free subgroup or G is a metabelian unipotent algebraic group.

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Ivan Arzhantsev, Mikhail Zaidenberg. 2021-11-12. Tits-type alternative for certain groups acting on algebraic surfaces. https://doi.org/10.1090/proc%2F16324

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