On cube and Cremona rigidity for higher-rank lattices

For irreducible lattices in semisimple Lie groups of real rank at least $2$, we prove a cohomological vanishing result implying that any action on a CAT(0) cube complex fixes a vertex whenever every hyperplane stabilizer is solvable. As an application, we prove regularizability for actions of all higher-rank lattices by birational transformations on projective surfaces. We first use superrigidity for actions on infinite-dimensional real hyperbolic spaces to reduce to the de Jonqui\`eres group, and then apply our fixed-point theorem to the Jonqui\`eres complex. Our proof bypasses the direct use of property FW.