Mapping Class Groups and Interpolating Complexes:Rank

A family of interpolating graphs C(S, ξ) of complexity ξ is constructed for a surface S and -2 ≤ ξ ≤ ξ(S). For ξ =-2,-1, ξ(S)-1 these specialize to graphs quasi-isometric to the marking graph, the pants graph and the curve graph respectively. We generalize the notion of a hierarchy and Theorems of Brock–Farb and Behrstock–Minsky to show that the rank of C(S, ξ) is rξ, the largest number of disjoint copies of subsurfaces of complexity greater than ξ that may be embedded in S. The interpolating graphs C(S,ξ) interpolate between the pants graph and the curve graph.