# Relatively hyperbolic groups with free abelian second cohomology

## TL;DR

This paper investigates the second cohomology of relatively hyperbolic groups, showing it is free abelian under certain boundary and subgroup conditions, and explores the topological properties of the associated cusped space.

## Contribution

It establishes conditions under which the second cohomology of relatively hyperbolic groups is free abelian, extending previous results and analyzing the cusped space topology.

## Key findings

- Second cohomology $H^2(G,\mathbb{Z}G)$ is free abelian under specified conditions.
- The cusped space has semistable fundamental group at infinity.
- Local connectivity of the boundary is crucial for the main results.

## 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.