Families of commuting automorphisms, and a characterization of the affine space

TL;DR

This paper demonstrates that the structure of an affine space can be uniquely identified by its automorphism group, focusing on commutative subgroups within automorphisms of affine varieties.

Contribution

It provides a new characterization of affine spaces based solely on the group structure of their automorphisms, emphasizing the role of commuting automorphisms.

Findings

Affine space is uniquely determined by its automorphism group

Commutative subgroups of automorphisms are key to the characterization

The structure of automorphism groups reflects the geometry of affine varieties

Abstract

In this paper we show that an affine space is determined by the abstract group structure of its group of regular automorphisms in the category of connected affine varieties. To prove this we study commutative subgroups of the group of automorphisms of affine varieties.