Mapping Class Groups and Interpolating Complexes: Rank

TL;DR

This paper introduces a family of interpolating graphs for surfaces that unify known complexes like the marking, pants, and curve graphs, and determines their rank based on surface subsurfaces.

Contribution

It generalizes existing theorems to define the rank of these interpolating graphs in terms of disjoint subsurfaces of certain complexities.

Findings

The rank of $\\calC (S, \\xi)$ is $r_\xi$, the maximum number of disjoint subsurfaces with complexity > \xi.

Interpolating graphs connect the marking, pants, and curve graphs as special cases.

The rank characterization extends previous results to a broader class of complexes.

Abstract

A family of interpolating graphs $\calC (S, \xi)$ of complexity $\xi$ is constructed for a surface $S$ and $-2 \leq \xi \leq \xi (S)$. 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 $\calC (S, \xi)$ 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 $\calC (S, \xi)$ interpolate between the pants graph and the curve graph.