Birational invariants of volume preserving maps

We study the group of birational automorphisms of the $n$-dimensional projective space that preserve the standard torus invariant volume form with logarithmic poles. We prove that this group is not generated by pseudo-regularizable maps for $n\ge 4$ over $\mathbb{C}$, and for $n\ge 3$ over number fields. As a corollary, we show that this group is not simple in these cases.