Strong Banach Property (T) for Simple Algebraic Groups of Higher Rank

TL;DR

This paper extends the proof of strong Banach property (T) from $SL_3$ to $Sp_4$ and higher rank algebraic groups, impacting the theory of expanders and fixed point properties in Banach spaces.

Contribution

It generalizes strong Banach property (T) to a broader class of algebraic groups with higher split rank, building on previous work for $SL_3$.

Findings

Families of expanders from such groups do not embed into Banach spaces of type > 1.

Any isometric affine action on Banach spaces of type > 1 has a fixed point.

The results apply to groups with $F$-split rank ≥ 2.

Abstract

In [Laf08], [Laf09], Vincent Lafforgue proved strong Banach property (T) for $SL_3$ over a non archimedean local field $F.$ In this paper, we extend his results to $Sp_4$ and therefore to any connected almost $F$-simple algebraic group with $F$-split rank $\geq 2.$ As applications, the family of expanders constructed from any lattice of such a group do not admit a uniform embedding into any Banach space of type $>1,$ and any isometric affine action of such a group, or its cocompact lattice, on a Banach space of type $>1$ has a fixed point.