The Cremona group is compactly presentable

TL;DR

This paper proves that the Cremona group, a fundamental object in algebraic geometry, has a compact presentation by expressing it as a generalized amalgamated product of specific algebraic subgroups with a single defining relation.

Contribution

It establishes the first known compact presentation of the Cremona group, revealing its structure as a generalized amalgamated product divided by one relation.

Findings

Cremona group is a generalized amalgamated product of three algebraic subgroups.

The group admits a compact presentation with a single relation.

The structure simplifies understanding of Cremona group's algebraic properties.

Abstract

This article shows that the Cremona group is compactly presentable. To prove this we show that it is a generalised amalgamated product of three of its algebraic subgroups (automorphisms of the plane and Hirzebruch surfaces) divided by one relation.