Automorphisms of algebraic varieties and infinite transitivity

TL;DR

This paper surveys recent advances in the study of automorphism groups of affine algebraic varieties, focusing on infinite transitivity and its algebraic and geometric implications.

Contribution

It provides a comprehensive overview of conditions for infinite transitivity and explores cases where automorphism groups are generated by finitely many one-parameter subgroups.

Findings

Infinite transitivity implies flexibility of affine varieties.

Many classes of varieties exhibit infinite transitivity.

Automorphism groups generated by finitely many one-parameter subgroups can also be infinitely transitive.

Abstract

We survey recent results on multiple transitivity of automorphism groups of affine algebraic varieties. We consider the property of infinite transitivity of the special automorphism group, which is equivalent to flexibility of the corresponding affine variety. These properties have important algebraic and geometric consequences. At the same time they are fulfilled for wide classes of varieties. Also we study situations where infinite transitivity takes place for automorphism groups generated by finitely many one-parameter subgroups. In the appendices to the paper, the results on infinitely transitive actions in complex analysis and in combinatorial group theory are discussed.