Finite subgroups of Cremona groups and representation dimension

The Cremona group of rank n is the group of birational automorphisms of n-dimensional projective space.  Alternatively, the Cremona group of rank n is the group of automorphisms of a purely transcendental extension.  Cremona groups are infamously large.  In particular, even the plane Cremona group cannot be embedded into a linear algebraic group.  Their finite subgroups are much more manageable, but still not completely understood even in the rank 2 case over the complex numbers.  However, lacking a complete classification, one may attempt to find bounds for their complexity.  Over a number field, one can consider the orders of the finite subgroups.  Over general fields, there is the Jordan constant.  I consider the minimal dimension of a faithful representation.  This is joint work with B. Heath and C. Urech.