Simple relations in the Cremona group

TL;DR

This paper provides a straightforward presentation of the plane Cremona group using generators and a single relation, simplifying its algebraic structure understanding.

Contribution

It introduces a simple set of generators and a unique relation that completely describe the Cremona group of the plane.

Findings

Cremona group is an amalgamated product of specific subgroups.

The group is characterized by a single relation involving key elements.

Simplifies the algebraic understanding of the Cremona group.

Abstract

We give a simple set of generators and relations for the Cremona group of the plane. Namely, we show that the Cremona group is the amalgamated product of the de Jonqui\`eres group with the group of automorphisms of the plane, divided by one relation which is $\sigma\tau=\tau\sigma$, where $\tau=(x:y:z)\mapsto (y:x:z)$ and $\sigma=(x:y:z)\dasharrow (yz:xz:xy)$.