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Suppose $G$ is a 1-ended finitely presented group that is hyperbolic relativeto $\mathcal P$ a finite collection of 1-ended finitely presented propersubgroups of $G$. Our main theorem states that if the boundary $\partial(G,{\mathcal P})$ is locally connected and the second cohomology group$H^2(P,\mathbb ZP)$ is free abelian for each $P\in \mathcal P$, then$H^2(G,\mathbb ZG)$ is free abelian. When $G$ is 1-ended it is conjectured that$\partial (G,\mathcal P)$ is always locally connected. Under mild conditions on$G$ and the members of $\mathcal P$ the 1-ended and local connectivityhypotheses can be eliminated and the same conclusion is obtained. When $G$ andeach member of $\mathcal P$ is 1-ended and $\partial (G,\mathcal P)$ is locallyconnected, we prove that the "Cusped Space" for this pair has semistablefundamental group at $\infty$. This provides a starting point in our proof ofthe main theorem.

Relatively hyperbolic groups with free abelian second cohomology