Sphere boundaries of hyperbolic groups

Abstract

We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-1 surface subgroups has $\partial G \cong \mathbb{S}^2$. Combined with a result of Markovic, our result gives a new characterization of virtually fundamental groups of hyperbolic 3-manifolds. Keywords: CAT(0) cube complexes, Hyperbolic groups, hyperbolic 3-manifolds.