Relatively hyperbolic groups with fixed peripherals
  We build quasi--isometry invariants of relatively hyperbolic groups which detect the hyperbolic parts of the group; these are variations of the stable dimension constructions previously introduced by the authors.   We prove that, given any finite collection of finitely generated groups $\mathcal{H}$ each of which either has finite stable dimension or is non-relatively hyperbolic, there exist infinitely many quasi--isometry types of one--ended groups which are hyperbolic relative to $\mathcal{H}$.   The groups are constructed using small cancellation theory over free products. 
Matthew Cordes and David Hume
math.GR math.GT math.MG