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, $\mathcal{A}\mathcal{M}(S)$. Adapting work of Masur-Minsky, we prove that $\mathcal{A}\mathcal{M}(S)$ 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 $\mathcal{A}\mathcal{M}(S)$ 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.