  Pseudo-automorphisms are birational transformations acting as regular
automorphisms in codimension 1. We import ideas from geometric group theory to
prove that a group of birational transformations that satisfies a fixed point
property on CAT(0) cubical complexes, for example a discrete countable group
with Kazhdan Property (T), is birationally conjugate to a group acting by
pseudo-automorphisms on some non-empty Zariski-open subset. We apply this
argument to classify groups of birational transformations of surfaces with this
fixed point property up to birational conjugacy.
