GF(2)-expansions

Abstract

Let Γ be a finite geometry of rank n ≥ 2 with a selected type of elements, called'points'. Let m be the number of'points' of Γ. Under some mild hypotheses on Γ we can consider an affine expansion of Γ to AG(m,2). We prove that the geometries obtained by applying this construction to matroids are simply connected. Then we exploit this result to study universal covers of certain geometries arising from hyperbolic quadrics and symplectic varieties over GF.