Normal subgroups in the Cremona group (long version)

TL;DR

This paper demonstrates that the Cremona group of birational transformations of the projective plane over an algebraically closed field is not simple, using hyperbolic geometry, geometric group theory, and algebraic geometry to identify non-trivial normal subgroups.

Contribution

It provides the first proof that the Cremona group is not simple, employing a novel combination of geometric and algebraic methods.

Findings

Cremona group is not simple

Existence of non-trivial normal subgroups in Cremona group

Application of hyperbolic geometry to algebraic group theory

Abstract

Let k be an algebraically closed field. We show that the Cremona group of all birational transformations of the projective plane P^2 over k is not a simple group. The strategy makes use of hyperbolic geometry, geometric group theory, and algebraic geometry to produce elements in the Cremona group that generate non trivial normal subgroups.