Tits-type alternative for certain groups acting on algebraic surfaces

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 X is a toric affine variety and G ⊂ Aut ⁡ ( X ) G\subset \operatorname {Aut}(X) is a subgroup generated by a finite set of unipotent subgroups normalized by the acting torus then either G G contains a nonabelian free subgroup or G G is a unipotent affine algebraic group. In the present paper we extend the latter result to any group G G of automorphisms of a complex affine surface generated by a finite collection of unipotent algebraic subgroups. It occurs that either G G contains a nonabelian free subgroup or G G is a metabelian unipotent algebraic group.