Quasi-isometric rigidity of Fuchsian buildings

We show that if a homeomorphism between the ideal boundaries of two Fuchsian buildings preserves the combinatorial cross ratio almost everywhere, then it extends to an isomorphism between the Fuchsian buildings. Together with the results of Bourdon–Pajot and Kleiner, it implies the quasi-isometric rigidity for Fuchsian buildings: any quasi-isometry between two Fuchsian buildings that admit cocompact lattices must lie at a finite distance from an isomorphism.