Divergence groups have the Bowen property

We show that every deformation of a divergence-type Fuchsian group has a limit set which is either a circle or has dimension strictly larger than 1. This is known to be false for all convergence-type groups, and hence solves Bowen's problem in the general case. The proof uses a theorem of Dennis Sullivan's about convex hulls in hyperbolic 3-space and we give a new, simpler proof of this result as well.