The augmented marking complex of a surface

Abstract

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, $\mathcal{AM}(S)$. Adapting work of Masur-Minsky, we prove that $\mathcal{AM}(S)$ is quasiisometric to Teichm\"uller space with the Teichm\"uller metric. A similar construction was independently discovered by Eskin-Masur-Rafi. We also completely integrate the Masur-Minsky hierarchy machinery to $\mathcal{AM}(S)$ to build flexible families of uniform quasigeodesics in Teichm\"uller space. As an application, we give a new proof of Rafi's distance formula for the Teichm\"uller metric.