Combinatorics of affine birational maps

TL;DR

This paper investigates the structure of the group of unimodular automorphisms of complex n-space, aiming to develop methods to analyze their properties, such as non-simplicity for dimensions three and higher.

Contribution

It introduces a new approach to study the combinatorics of affine birational maps, specifically focusing on the group of unimodular automorphisms.

Findings

Proves the non-simplicity of the automorphism group for all n ≥ 3

Develops a systematic method for analyzing affine birational maps

Provides a framework for future generalizations

Abstract

The main object of study of the present paper is the group $\au_n$ of \emph{unimodular automorphisms} of $\com^n$. Taking $\au_n$ as a working example, our intention was to develop an approach (or rather an edifice) which allows one to prove, for instance, the non-simplicity of $\au_n$ for all $n \geq 3$. More systematic and, perhaps, general exposition will appear elsewhere.