Commensurating actions of birational groups and groups of pseudo-automorphisms

Preliminary version. 41 pages, 4 figuresInternational audiencePseudo-automorphisms are birational transformations acting as regular automorphisms in codimension 1. We import ideas from geometric group theory to study groups of birational transformations, and prove that a group of birational transformations that satisfies a fixed point property on CAT(0) cubical complexes is birationally conjugate to a group acting by pseudo-automorphisms on a 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