Infinite transitivity and special automorphisms

TL;DR

This paper investigates conditions under which the automorphism group of a quasiaffine variety acts infinitely transitively, establishing that certain group actions imply infinite transitivity and exploring the implications of 2-transitivity.

Contribution

It proves that for varieties with a nontrivial bG_a or bG_m action, automorphism groups acting infinitely transitively are characterized by the special automorphism group, and 2-transitivity implies infinite transitivity.

Findings

Special automorphism group acting transitively implies infinite transitivity.

Presence of bG_a or bG_m actions characterizes infinite transitivity.

2-transitivity of automorphism group implies infinite transitivity.

Abstract

It is known that if the special automorphism group $\text{SAut}(X)$ of a quasiaffine variety $X$ of dimension at least $2$ acts transitively on $X$, then this action is infinitely transitive. In this paper we address the question whether this is the only possibility for the automorphism group $\text{Aut}(X)$ to act infinitely transitively on $X$. We show that this is the case provided $X$ admits a nontrivial $\mathbb{G}_a$- or $\mathbb{G}_m$-action. Moreover, 2-transitivity of the automorphism group implies infinite transitivity.