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

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In [Laf08], [Laf09], Vincent Lafforgue proved strong Banach property (T) for SL₃ over a non archimedean local field F. In this paper, we extend his results to Sp₄ and therefore to any connected almost F-simple algebraic group with F-split rank ≥ 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.