Relatively hyperbolic groups with free abelian second cohomology

Abstract

Suppose $G$ is a 1-ended finitely presented group that is hyperbolic relative to $\mathcal P$ a finite collection of 1-ended finitely presented proper subgroups 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 connectivity hypotheses can be eliminated and the same conclusion is obtained. When $G$ and each member of $\mathcal P$ is 1-ended and $\partial (G,\mathcal P)$ is locally connected, we prove that the "Cusped Space" for this pair has semistable fundamental group at $\infty$. This provides a starting point in our proof of the main theorem.