Local indicability and relative presentations of groups

A relative presentation is a triple where A is a group, X is a set, and R is a set of words in the free product A*F(X) where F(X) is the free group with basis X. Under certain hypotheses on the relative presentation , we show that (1) the group presented by is locally indicable; (2) the pre-aspherical model for is aspherical; (3) the Freiheitssatz holds for . The result has applications in the computation of cohomology of groups and the field of equations over groups.