Strong Property (T), weak amenability and ${\ell }^p$-cohomology in ${\tilde{A}}_2$-buildings

Abstract

We prove that cocompact (and more generally: undistorted) lattices on ${\tilde{A}}_2$-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 ${\ell }^p$-cohomology of an ${\tilde{A}}_2$-building vanishes for any finite $p$. Second, we show that the non-commutative $L^p$-space for $p$ not in $[\frac{4}{3},4]$ and the reduced $C^{\ast }$-algebra associated to an ${\tilde{A}}_2$-lattice do not have the operator space approximation property and, consequently, that the lattice is not weakly amenable.

arXiv commentv1: 68 pages, 6 figures; v2: 79 pages, many improvements in the presentation. To appear in Ann. Sci. École Norm. Sup

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