Is the Cremona group compactly presented?

The Cremona group is the group of birational transformations of the projective space over a field k of dimension n.
For local fields k, there exists a topology on the Cremona group that makes it a Hausdorff topological group and which restricted to the subgroup PGL(k,n+1) of linear transformations is exactly the Euclidean topology on PGL(k,n+1). Naturally, we ask: What are the group’s topological properties?
Endowed with this topology, the Cremona group is not locally compact if n>1, but it is the inductive limit of locally compact closed sets. I will discuss which Cremona groups are compactly presented (i.e. have a compact generating set and generating relations of bounded length) and why, despite not being locally compact.