Garland’s method with Banach coefficients

We prove a Banach version of Garland's method proving vanishing cohomology for groups acting on simplicial complexes. The novelty this new is that our condition applies to every reflexive space. This allows us deduce several criteria group with coefficients in classes spaces (uniformly curved spaces, Hilbertian and $L^p$ spaces). Using these criteria, we improve recent results regarding fixed point theorems random the triangular model give sharp lower bound conformal dimension boundary such groups. Also, derive stability respect p-Schatten norms.