Smooth Non-Homogeneous Gizatullin Surfaces

TL;DR

This paper investigates the structure of automorphism groups of Gizatullin surfaces, showing new examples of non-homogeneous surfaces with large, complex automorphism groups and less restrictive conditions than previously known.

Contribution

It extends the class of known Gizatullin surfaces with non-transitive automorphism groups and demonstrates the complexity of their automorphism group structures.

Findings

Existence of smooth affine surfaces with large automorphism groups

Examples of non-homogeneous Gizatullin surfaces under weaker conditions

Automorphism groups contain free groups over uncountable sets

Abstract

Quasi-homogeneous surfaces, or Gizatullin surfaces, are normal affine surfaces such that there exists an open orbit of the automorphism group with a finite complement. If the action of the automorphism group is transitive, the surface is called homogeneous. Examples of non-homogeneous Gizatullin surfaces were constructed in [Ko], but on more restricted conditions. We show that a similar result holds under less constrained assumptions. Moreover, we exhibit examples of smooth affine surfaces with a non-transitive action of the automorphism group whereas the automorphism group is huge. This means that it is not generated by a countable set of algebraic subgroups and that its quotient by the (normal) subgroup generated by all algebraic subgroups contains a free group over an uncountable set of generators.