On automorphism groups of affine surfaces

TL;DR

This survey explores the structure and classification of automorphism groups of affine algebraic surfaces, emphasizing their infinite-dimensional nature and group-theoretic properties, while highlighting new results and open problems.

Contribution

It offers a comprehensive overview of automorphism groups of affine surfaces, introduces new results, and discusses various classification approaches with a focus on group-theoretic methods.

Findings

Automorphism groups are infinite-dimensional ind-groups.

Classification methods include combinatorial group theory approaches.

Several open problems in the structure and classification of these groups are identified.

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.