Tits-type alternative for certain groups acting on algebraic surfaces

TL;DR

This paper extends a Tits-type alternative to groups of automorphisms of complex affine surfaces, showing they either contain a free subgroup or are a specific type of unipotent algebraic group.

Contribution

It generalizes previous results to broader classes of automorphism groups acting on complex affine surfaces, establishing a dichotomy involving free subgroups and unipotent algebraic groups.

Findings

Groups contain a nonabelian free subgroup or are unipotent algebraic groups.

Extension of Tits-type alternative to automorphism groups of affine surfaces.

Identification of conditions under which groups are metabelian unipotent.

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.