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, $\mathcal{A}\mathcal{M}(S)$. Adapting work of Masur-Minsky, we prove that $\mathcal{A}\mathcal{M}(S)$ is quasiisometric to Teichmüller space with the Teichmüller 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 Teichmüller space. As an application, we give a new proof of Rafi's distance formula for the Teichmüller metric.

30 pages; significantly rewritten to strengthen main constructions

References in corpus (2)

• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• The asymptotic geometry of the Teichmüller metric: Dimension and rank

Cited by in corpus (7)

• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Boundaries and automorphisms of hierarchically hyperbolic spaces
• Deviation inequalities and CLT for random walks on acylindrically hyperbolic groups
• Large scale rank of Teichmuller space
• Stable cubulations, bicombings, and barycenters
• Bounded Cohomology of Finitely Generated Kleinian Groups
• Extensions of multicurve stabilizers are hierarchically hyperbolic