Banach property (T) for $\mathrm{S}{\mathrm{L}}_{\mathrm{n}}(\mathbb{Z})$ and its applications

Abstract

We prove that a large family of higher rank simple Lie groups (including $\mathrm{S}{\mathrm{L}}_{\mathrm{n}}(\mathbb{R})$ for $n\ge 3$) 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 $n\ge 4$, the group $\mathrm{S}{\mathrm{L}}_{\mathrm{n}}(\mathbb{R})$ 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 $\mathrm{S}{\mathrm{L}}_{\mathrm{n}}(\mathbb{Z}/m\mathbb{Z})$ for a fixed $n\ge 3$ and $m$ tending to infinity) are super-expanders. All of our results stem from proving Banach property (T) for $\mathrm{S}{\mathrm{L}}_3(\mathbb{Z})$. Our method of proof for $\mathrm{S}{\mathrm{L}}_3(\mathbb{Z})$ relies on a novel proof for relative Banach property (T) for the uni-triangular subgroup of $\mathrm{S}{\mathrm{L}}_3(\mathbb{Z})$. This proof of relative property (T) is new even in the classical Hilbert setting and is interesting in its own right.

Corrected typos. 35 pages. To appear in Inventiones Mathematicae

References in corpus (4)

• On strong property (T) and fixed point properties for Lie groups
• Superexpanders from group actions on compact manifolds
• Strong property (T) for higher rank lattices
• Super-expanders and warped cones