Structure of connected nested automorphism groups

TL;DR

This paper characterizes the structure of maximal unipotent and commutative unipotent subgroups of automorphism groups of affine varieties, establishing conditions for their closure and providing explicit constructions, thus advancing understanding of automorphism group structures.

Contribution

It introduces a detailed description of maximal unipotent subgroups of automorphism groups of affine varieties, extending recent results and answering open questions about their closure properties.

Findings

Maximal unipotent subgroups are nested and analogous to triangular automorphisms.

A unipotent subgroup is closed if and only if it is nested.

Connected nested unipotent subgroups are necessarily closed.

Abstract

In this article, we describe the maximal unipotent subgroups of $\mathrm{Aut}(X)$, where $X$ is an affine algebraic variety. Every subgroup of this type has a structure analogous to that of the group of triangular automorphisms of $\mathbb{A}^n$. In particular, it is nested, that is, a countable increasing union of algebraic subgroups. We show that a subgroup $G\subset\mathrm{Aut}(X)$ consisting of unipotent elements is closed if and only if it is nested. This implies that a connected nested subgroup of $\mathrm{Aut}(X)$ is closed, thus answering a question posed by Kraft and Zaidenberg (2022). We also extend the recent description of maximal commutative unipotent subgroups of $\mathrm{Aut}(X)$ due to Regeta and van Santen (2024), by providing a direct construction of such subgroups within our approach.