Mapping Class Groups and Interpolating Complexes: Rank

A family of interpolating graphs $\text{calC}(S,\xi )$ of complexity $\xi$ is constructed for a surface $S$ and $-2\le \xi \le \xi (S)$. For $\xi = -2, -1, \xi (S) -1$ these specialise to graphs quasi-isometric to the marking graph, the pants graph and the curve graph respectively. We generalise Theorems of Brock-Farb and Behrstock-Minsky to show that the rank of $\text{calC}(S,\xi )$ is $r_{\xi }$, the largest number of disjoint copies of subsurfaces of complexity greater than $\xi$ that may be embedded in $S$. The interpolating graphs $\text{calC}(S,\xi )$ interpolate between the pants graph and the curve graph.