Birational Invariants of volume-preserving maps

One of the main problems of birational geometry is the classification of algebraic varieties up to birational equivalence. Refining this problem, one can classify algebraic varieties with additional structure, for example, by considering varieties with a fixed (meromorphic) volume form. In this case, it is natural to consider volume forms that have poles of at most first order. The group of equivalence classes of varieties with such a form is called the Burnside group. This group is good because some natural invariants of birational maps preserving the volume form on a given variety take values in it. We will define and study these invariants (sometimes called "motivic invariants") for groups of birational automorphisms of a projective space with a "standard" toric-invariant form. We will show that such groups are not simple in any dimension starting from four, and also that they cannot be generated by pseudo-regularizable elements. This result can be seen as a generalization of a similar theorem for the classical Cremona group, that is, the group of birational automorphisms of the projective space.