Mapping Class Groups and Interpolating Complexes: Rank

Original · EN

A family of interpolating graphs (S, ξ) of complexity ξ is constructed for a surface S and -2 ≤ ξ≤ 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 (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 (S, ξ) interpolate between the pants graph and the curve graph.