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

TL;DR

This paper establishes Banach property (T) for higher rank simple Lie groups and their lattices, leading to fixed point properties and super-expander results, including a novel proof for $ m SL_3 (bZ)$.

Contribution

It proves Banach property (T) for $ m SL_n (bR)$ and lattices, and introduces a new proof for relative property (T) for the unipotent subgroup of $ m SL_3 (bZ)$.

Findings

Higher rank simple Lie groups have Banach property (T) for all super-reflexive Banach spaces.

$ m SL_n (bR)$ and its lattices have the Banach fixed point property for $n extgreater 3$.

Cayley graphs of $ m SL_{n} (bZ/mbZ)$ are super-expanders for fixed $n extgreater 2$.

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.