Banach property (T) for $\rm SL_n (\mathbb{Z})$ and its applications

Abstract: We prove that a large family of higher rank simple Lie groups (including $\rm SL_n (\mathbb{R})$ for $n \geq 3$) 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 $n \geq 4$, the group $\rm SL_n (\mathbb{R})$ 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 $\rm SL_{n} (\mathbb{Z} / m \mathbb{Z} )$ for a fixed $n \geq 3$ and $m$ tending to infinity) are super-expanders. All of our results stem from proving Banach property (T) for $\rm SL_3 (\mathbb{Z})$. Our method of proof for $\rm SL_3 (\mathbb{Z})$ relies on a novel proof for relative Banach property (T) for the uni-triangular subgroup of $\rm SL_3 (\mathbb{Z})$. This proof of relative property (T) is new even in the classical Hilbert setting and is interesting in its own right.