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

In [Laf08], [Laf09], Vincent Lafforgue proved strong Banach property (T) for $S{L}_3$ over a non archimedean local field $F\mathrm{.}$ In this paper, we extend his results to $S{p}_4$ and therefore to any connected almost $F$-simple algebraic group with $F$-split rank $\ge 2.$ 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.