Tits-type alternative for certain groups acting on algebraic surfaces

Arzhantsev, Ivan; Zaidenberg, Mikhail

doi:10.1090/proc/16324

Tits-type alternative for certain groups acting on algebraic surfaces
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by Ivan Arzhantsev and Mikhail Zaidenberg

Proc. Amer. Math. Soc. 151 (2023), 2813-2829

DOI: https://doi.org/10.1090/proc/16324

Published electronically: March 30, 2023

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Abstract:

According to a theorem of Cantat [Ann. of Math. (2) 174 (2011), pp. 299–340] and Urech [J. Reine Angew. Math. 770 (2021), pp. 27–57], an analog of the classical Tits alternative holds for the group of birational automorphisms of a compact complex Kähler surface. We established in Arzhantsev and Zaidenberg [Int. Math. Res. Not. IMRN 2022 (2022), pp. 8162–8195] the following Tits-type alternative: if $X$ is a toric affine variety and $G\subset \operatorname {Aut}(X)$ is a subgroup 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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