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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. [ABSTRACT FROM AUTHOR]

We give a presentation of the plane Cremona group over an algebraically closed field with respect to the generators given by the theorem of Noether and Castelnuovo. This presentation is particularly simple and can be used for explicit calculations. [ABSTRACT FROM AUTHOR]