Strong property (T) for higher-rank simple Lie groups: Figure 1.

Abstract

We prove that connected higher-rank simple Lie groups have Lafforgue's strong property (T) with respect to a certain class of Banach spaces E 10 containing many classical superreflexive spaces and some non-reflexive spaces as well. This generalizes the result of Lafforgue asserting that SL ( 3 , R ) has strong property (T) with respect to Hilbert spaces and the more recent result of the second-named author asserting that SL ( 3 , R ) has strong property (T) with respect to a certain larger class of Banach spaces. For the generalization to higher-rank groups, it is sufficient to prove strong property (T) for Sp ( 2 , R ) and its universal covering group. As consequences of our main result, it follows that for X ∈ E 10 , connected higher-rank simple Lie groups and their lattices have property ( F X ) of Bader, Furman, Gelander and Monod, and that the expanders constructed from a lattice in a connected higher-rank simple Lie group do not admit a coarse embedding into X.