Strong Banach property (T), after Vincent Lafforgue

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 $S{L}_3$ 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 $s{l}_3(F)$ 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.

This text is an improved version of the author's master thesis

References in corpus (1)

• On fixed points and uniformly convex spaces