Relatively hyperbolic groups with free abelian second cohomology

Suppose G is a finitely presented group that hyperbolic relative to P finite collection of proper subgroups G. Our main theorem states if the second cohomology H2(P,ZP) free abelian for each P?P, then H2(G,ZG) abelian. The problem reduces case when and members are 1-ended. In case, we prove “Cusped Space” this pair has semistable fundamental at ?. This provides starting point in our proof theorem. Finally simplicial approximation result maps [0,?)×[0,1]. cusped space primary components Both have applications beyond article.