# When is the automorphism group of an affine variety nested?

## TL;DR

This paper characterizes when the subgroup generated by connected algebraic subgroups of automorphisms of an affine variety is nested, linking this property to the commutativity of all additive group actions on the variety.

## Contribution

It establishes a precise criterion for the nestedness of the automorphism subgroup based on the commutativity of all -actions, and describes its structure.

## Key findings

- -actions on the variety commute iff the automorphism subgroup is nested.
- The subgroup -automorphisms form a direct limit of algebraic subgroups.
- The structure of -automorphism subgroup is explicitly described.

## Abstract

For an affine algebraic variety $X$, we study the subgroup $\mathrm{Aut}_{\text{alg}}(X)$ of the group of regular automorphisms $\mathrm{Aut}(X)$ of $X$ generated by all the connected algebraic subgroups. We prove that $\mathrm{Aut}_{\text{alg}}(X)$ is nested, i.e., is a direct limit of algebraic subgroups of $\mathrm{Aut}(X)$, if and only if all the $\mathbb{G}_a$-actions on $X$ commute. Moreover, we describe the structure of such a group $\mathrm{Aut}_{\text{alg}}(X)$.