Conjugacy classes of special automorphisms of the affine spaces

TL;DR

This paper investigates the conjugacy classes of special automorphisms of affine spaces, revealing that unlike polynomial automorphisms of the plane, their conjugacy classes are not necessarily closed, which has implications for the group's structure.

Contribution

It demonstrates that conjugacy class closure properties differ between polynomial automorphisms and special automorphisms, advancing understanding of the group's algebraic structure.

Findings

Conjugacy classes of special automorphisms are not always closed.

Diagonalisability characterizes closed conjugacy classes in polynomial automorphisms.

Results suggest the special automorphism group may not be simple.

Abstract

In the group of polynomial automorphisms of the plane, the conjugacy class of an element is closed if and only if the element is diagonalisable. In this article, we show that this does not hold for the group of special automorphisms, giving then a first step towards the direction of showing that this group is not simple, as an infinite-dimensional algebraic group.