Tits-type alternative for certain groups acting on algebraic surfaces

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.

16 pages; extended by an alternative, short proof of the main theorem valid over any algebraically closed field of characteristic zero. To appear in: Proc. Amer. Math. Soc

References in corpus (6)

• Tits type alternative for groups acting on toric affine varieties
• Algebraic subgroups of the plane Cremona group over a perfect field
• Nonlinearity of some subgroups of the planar Cremona group
• Some finiteness results on triangular automorphisms
• Borel subgroups of the plane Cremona group
• Finite groups of birational transformations