Strong Property (T), weak amenability and ᵖ-cohomology in A₂-buildings

We prove that cocompact (and more generally: undistorted) lattices on A₂-buildings satisfy Lafforgue's strong property (T), thus exhibiting the first examples that are not related to algebraic groups over local fields. Our methods also give two further results. First, we show that the first ᵖ-cohomology of an A₂-building vanishes for any finite p. Second, we show that the non-commutative Lᵖ-space for p not in 4 3, 4 and the reduced C^*-algebra associated to an A₂-lattice do not have the operator space approximation property and, consequently, that the lattice is not weakly amenable.