On automorphism groups of affine surfaces
Sergei Kovalenko, Alexander Perepechko, Mikhail Zaidenberg
TL;DR
The paper surveys automorphism groups of affine surfaces, emphasizing their infinite-dimensional, ind-group nature and organizing results via ML-invariants, rank, and group-theoretic frameworks. It develops and applies the language of ind-groups, nested ind-groups, amalgams, and bearable groups to classify automorphism groups and algebraic actions, including detailed treatments of Gizatullin and G_m-surface cases. Key contributions include clarifying ML$_0$/ML$_1$/ML$_2$ distinctions, presenting Dolgachev–Pinkham–Demazure (DPD) descriptions for ${ m G}_m$-surfaces, and providing amalgam decompositions and one-parameter subgroup classifications for notable surface classes. The work highlights open problems, such as the full structure of automorphism groups for Gizatullin surfaces and the bearability of automorphism groups in broader families, with implications for understanding symmetries of affine algebraic surfaces in both algebraic and combinatorial terms.
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.