Near coverings and cosystolic expansion

Let X, Y be simplicial complexes and let f:Y→X be a simplicial surjective map. We introduce a notion of deficiency of f, denoted by mf(Y), that measures the average local failure of f:Y→X to be a covering map. We show, roughly speaking, that if mf(Y) is small and and if the non-abelian cosystolic expansion of X is large, then f is close to a genuine covering map. Our main result is a lower bound on the 1-cosystolic expansion with G coefficients of geometric lattices, with an application to near coverings of the 2-dimensional spherical building A3(Fq).