Automorphisms of algebraic varieties and infinite transitivity

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.

arXiv comment39 pages

References in corpus (14)

• Flexible varieties and automorphism groups
• Affine T-varieties of complexity one and locally nilpotent derivations
• Flexibility of affine cones over del Pezzo surfaces of degree 4 and 5
• Ga-actions of fiber type on affine T-varieties
• Infinite transitivity on universal torsors
• The automorphism group of a rigid affine variety
• Infinite transitivity, finite generation, and Demazure roots
• Cylinders in singular del Pezzo surfaces
• Tits type alternative for groups acting on toric affine varieties
• Affine cones over cubic surfaces are flexible in codimension one
• A new notion of Tameness
• Lie algebras of vertical derivations on semiaffine varieties with torus actions
• Highly transitive actions of free products
• Transitivity degrees of countable groups and acylindrical hyperbolicity