The augmented marking complex of a surface

We build an augmentation of the Masur-Minsky marking complex by Groves-Manning combinatorial horoballs to obtain a graph we call the augmented marking complex, AMpSq. Adapting work of Masur-Minsky, we show this augmented marking complex is quasiisometric to Teichmüller space with the Teichmüller metric. A similar construction was independently discovered by Eskin-Masur-Rafi [EMR13]. We also completely integrate the Masur-Minsky hierarchy machinery to AMpSq to build flexible families of uniform quasigeodesics in Teichmüller space. As an application, we give a new proof of Rafi’s distance formula for T pSq with the Teichmüller metric. We have included an appendix in which we prove a number of facts about hierarchies that we hope will be of independent interest.