Generalised Magnus modules over the braid group

W.~Magnus' representations of submonoids $E \leq \mbox{End}(F)$ of the endomorphisms of a free group $F$ of finite rank are generalised by identifying them with the first homology group of $F$ with particular coefficient modules. By considering a suitable free resolution of the integers over the semidirect product of free groups, a class of representations of the braid group can be obtained on higher homology groups. The resolution shows that the holonomy representations of the braid group and of the Hecke algebra constructed topologically by R.~J.~Lawrence belong to this class.