Mapping Class Groups and Interpolating Complexes: Rank

A family of interpolating graphs $\calC (S, ξ)$ of complexity $ξ$ is constructed for a surface $S$ and $-2 \leq ξ\leq ξ(S)$. For $ξ= -2, -1, ξ(S) -1$ 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 $\calC (S, ξ)$ is $r_ξ$, the largest number of disjoint copies of subsurfaces of complexity greater than $ξ$ that may be embedded in $S$. The interpolating graphs $\calC (S, ξ)$ interpolate between the pants graph and the curve graph.

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