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

TL;DR

This paper demonstrates that certain lattices on $ ilde{A}_2$-buildings possess strong property (T), and explores their $ ext{L}^p$-cohomology and operator space properties, revealing new examples outside algebraic group contexts.

Contribution

It proves that cocompact lattices on $ ilde{A}_2$-buildings have strong property (T) and analyzes their $ ext{L}^p$-cohomology and operator space properties, providing novel examples beyond algebraic groups.

Findings

Cocompact lattices on $ ilde{A}_2$-buildings satisfy strong property (T).

First $ ext{L}^p$-cohomology vanishes for all finite p.

Non-commutative $ ext{L}^p$-spaces and reduced $C^*$-algebras lack the operator space approximation property.

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 $C^*$-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.