Strong property $(T)$ for higher-rank lattices

Abstract

We prove that every lattice in a product of higher-rank simple Lie groups or higher-rank simple algebraic groups over local fields has Vincent Lafforgue’s strong property $(T)$. Over non-Archimedean local fields, we also prove that they have strong Banach property $(T)$ with respect to all Banach spaces with non-trivial type, whereas in general we obtain such a result with additional hypotheses on the Banach spaces. The novelty is that we deal with non-co-compact lattices, such as $\mathop{\rm SL}_n(\mathbf{Z})$ for $n \geqslant 3$. To do so, we introduce a stronger form of strong property $(T)$ which allows us to deal with more general objects than group representations on Banach spaces that we call two-step representations, namely families indexed by a group of operators between different Banach spaces that we can compose only once. We prove that higher-rank groups have this property and that this property passes to undistorted lattices.