Strong Banach property (T), after Vincent Lafforgue

TL;DR

This paper proves that certain p-adic groups, including SL3 over non-archimedean fields, have strong Banach property (T), leading to fixed point properties and non-embeddability results for associated expanders.

Contribution

It extends the proof of strong Banach property (T) to SL3 over non-archimedean fields and applies this to derive fixed point and non-embeddability results for related groups and expanders.

Findings

SL3 over non-archimedean fields has strong Banach property (T)

Expanders from lattices of these groups do not embed into Banach spaces of type >1

Any affine isometric action of these groups on such Banach spaces has a fixed point

Abstract

This text takes, with more details and simplifying a proof in section 3, the parts of [Laf08] and [Laf09] treating p-adic groups. We prove that $SL_{3}$ over a non archimedian local field $F$ has strong Banach property (T). The applications are as follows: any connected almost $F$-simple algebraic group $G$ over $F$ whose Lie algebra contains $sl_3(F)$ has strong Banach property (T), the family of expanders constructed from a lattice of $G(F)$ do not embed uniformly in any Banach space of type $>1,$ and any isometric affine action of $G(F)$ on a Banach space of type $>1$ admits a fix point.