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, AM(S). Adapting work of Masur-Minsky, we show that this augmented marking complex 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 AM(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 T (S) 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.Theorem 1.1. The augmented marking complex, AM(S), is MCG(S)-equivariantly quasiisometric to T (S) in the Teichmüller metric.A large part of this paper is spent adapting the Masur-Minsky hierarchy machinery for M(S) and P(S) to AM(S). We use these augmented hierarchies for AM(S) to build families of uniform quasigeodesics called augmented hierarchy paths, and derive a version of Rafi's distance formula for the Teichmüller metric (Theorem 2.10), thereby completing the unification of the coarse geometries of MCG(S) and T (S) in the Weil-Petersson, and Teichmüller metrics by a common framework developed in [7,19,20,24,25]. In a recent paper, Eskin-Masur-Rafi [12] used AM(S) and augmented hierarchy paths, which they independently discovered,

Large-scale rank and rigidity of the Teichmüller metric

“…A direct proof for the augmented marking complex is given in [D1]. Given that these spaces are quasi-isometric [Ra,D1] these statements are equivalent, and both are equivalent to the corresponding statement for the decorated marking complex as we have described it. We note that another proof of this distance estimate can be found in [BeHS2].…”

“…The construction of the 'augmented marking graph' in [14] fits (more or less) into this picture. There, the marking graph M is taken to be the marking graph as defined in [23].…”

“…In this section, we describe a combinatorial model R = R(Σ) for T(Σ). It is a slight variation on the 'augmented marking complex' described in [28] and in [14]. In order to distinguish it, we will refer to the model described here as the 'decorated marking complex'.…”

“…In particular, we prove a number of results relating to the coarse rank of Teichmüller space, as well as quasi-isometric rigidity. Our starting point is a combinatorial model of Teichmüller space [Ra,D1], which we use to show that Teichmüller space admits a coarse median structure in the sense of [Bo1]. From this, a number of facts follow immediately, though others require additional work.…”

“…Note that the mapping class group Map(Σ) acts properly discontinuously on T(Σ). The properties we are mainly interested in here are quasi-isometry invariant, and hence for most of the paper we will be referring instead to the 'decorated marking graph', R(Σ), which is a slight variation on the 'augmented marking graph' of [14]. Both these spaces are equivariantly quasi-isometric to T(Σ).…”