COXETER COVERS OF THE CLASSICAL COXETER GROUPS

Let C(T) be a generalized Coxeter group, which has a natural map onto one of the classical Coxeter groups, either B(n) or D(n). Let C(Y) (T) be a natural quotient of C(T), and if C(T) is simply-laced (which means all the relations between the generators has order 2 or 3), C(Y) (T) is a generalized Coxeter group, too. Let At,n be a group which contains t Abelian groups generated by n elements. The main result in this paper is that C(Y) (T) is isomorphic to A(t,n) (sic) B(n) or A(t,n) (sic) D(n), depends on whether the signed graph T contains loops or not, or in other words C(T) is simply-laced or not, and t is the number of the cycles in T. This result extends the results of Rowen, Teicher and Vishne to generalized Coxeter groups which have a natural map onto one of the classical Coxeter groups.