\(GF(2)\)-expansions (Q1373238)

Let \(\Gamma\) be a finite geometry of rank \(n\geq 2\) with a selected type of elements called ``points''. Under mild hypotheses on \(\Gamma\) the authors consider an affine extension of \(\Gamma\) to \(AG(m,2)\), where \(m\) is the number of ``points'' in \(\Gamma\). The authors prove that geometries obtained by applying this construction to matroids are simply connected. This result is applied for the study of universal covers of certain geometries arising from hyperbolic quadrics and symplectic varieties over \(GF(q)\).