On automorphism groups of affine surfaces

Abstract

This is a survey on the automorphism groups in various classes of affine algebraic surfaces and the algebraic group actions on such surfaces. Being infinite-dimensional, these automorphism groups share some important features of algebraic groups. At the same time, they can be studied from the viewpoint of the combinatorial group theory, so we put a special accent on group-theoretical aspects (ind-groups, amalgams, etc.). We provide different approaches to classification, prove certain new results, and attract attention to several open problems.

Proposition 2.10 from the previous version (published in Algebraic Varieties and Automorphism Groups, ASPM 75) deleted. There is a mistake in the proof kindly indicated by J.-P. Furter; the validity of the result remains open. This does not affect the rest of the paper

References in corpus (7)

• Flexible varieties and automorphism groups
• Locally nilpotent derivations on affine surfaces with a $\C^*$-action
• Automorphisms of the plane preserving a curve
• Holomorphic automorphisms of Danielewski surfaces II -- structure of the overshear group
• Automorphisms of open surfaces with irreducible boundary
• Makar-Limanov invariant, Derksen invariant, flexible points
• The density property for Gizatullin surfaces of type $[[0,0,-r_2,-r_3]]$