Transitivity of automorphism groups of Gizatullin surfaces

TL;DR

This paper investigates the automorphism groups of a specific subclass of Gizatullin surfaces, showing they are generated by A1-fibration automorphisms and providing counterexamples to Gizatullin's conjecture.

Contribution

It demonstrates that certain Gizatullin surfaces have automorphism groups with non-transitive actions and describes these groups explicitly as amalgamated products.

Findings

Automorphism groups are generated by A1-fibration automorphisms.

Some Gizatullin surfaces have non-transitive automorphism group actions.

Explicit automorphism group structures as amalgamated products.

Abstract

We show that the automorphism group of a certain subclass of smooth Gizatullin surfaces with a distinguished and rigid extended divisor is generated by automorphisms of A1-fibrations. Moreover, such surfaces provide examples of smooth Gizatullin surfaces with a non-transitive action of the automorphism group. Thus, they represent counterexamples to Gizatullin's conjecture. For such surfaces we give explicit orbits of the natural action of the automorphism group in some special cases. Further, we present their automorphism groups as amalgamated products of two subgroups.