Large-scale geometry of graphs interpolating between curve graphs and pants graphs

We study two types of graphs interpolating between the curve graph and the pants graph from the viewpoint of large-scale geometry. One was introduced by Erlandsson and Fanoni, and the other by Mahan Mj. These graphs were developed independently in different contexts. In this paper, we provide explicit formulae for computing their quasi-flat ranks. These formulae depend on the genus and the number of boundary components of the underlying surface, as well as the interpolation parameter. We also classify geometries of the interpolating graphs into the hyperbolic, relatively hyperbolic, and thick cases. Our approach relies on the theory of twist-free graphs of multicurves, which is developed by Vokes and Russel.