Generators and relations of the Cremona group in dimension three

Determine a complete presentation—namely, a generating set together with defining relations—for the Cremona group Bir(P^3) of birational self-maps of complex projective 3-space. Establish whether Bir(P^3) admits a finite generating set and identify explicit relations among the generators to give a full group presentation.

Background

In dimension two, the classical Noether–Castelnuovo theorem states that the Cremona group Bir(P^2) is generated by PGL(3,C) together with the standard Cremona transformation. This gives a concrete and complete description of Bir(P^2) in terms of generators.

The authors note that extending such a presentation to higher dimensions is difficult. In particular, identifying generators and relations for Bir(P^3) remains unresolved. A solution would provide a foundational understanding of birational transformations in three dimensions, with implications across algebraic geometry and areas where birational dynamics play a role (e.g., integrable systems and algebraic dynamics).