Banach property (T) for SL _n(Z) and its applications

We prove that a large family of higher rank simple Lie groups (including SL _n(R) for n≥ 3) and their lattices have Banach property (T) with respect to all super-reflexive Banach spaces. Two consequences of this result are: First, we deduce Banach fixed point properties with respect to all super-reflexive Banach spaces for a large family of higher rank simple Lie groups. For example, we show that for every n≥ 4 , the group SL _n(R) and all its lattices have the Banach fixed point property with respect to all super-reflexive Banach spaces. Second, we settle a long standing open problem and show that the Margulis expanders (Cayley graphs of SL _n(Z/ mZ) for a fixed n≥ 3 and m tending to infinity) are super-expanders. All of our results stem from proving Banach property (T) for SL ₃(Z) . Our method of proof for SL ₃(Z) relies on a novel proof for relative Banach property (T) for the uni-triangular subgroup of SL ₃(Z) . This proof of relative property (T) is new even in the classical Hilbert setting and is interesting in its own right.