Mapping Class Groups and Interpolating Complexes: Rank

Abstract

A family of interpolating graphs $\calC (S, ξ)$ 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 $\calC (S, ξ)$ 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 $\calC (S, ξ)$ interpolate between the pants graph and the curve graph.

v2 Final version incorporating refree comments 16pgs no figs

References in corpus (2)

• Geometry of the Complex of Curves I: Hyperbolicity
• Asymptotic geometry of the mapping class group and Teichmueller space