Large scale rank of Teichmuller space

Abstract

Let X be quasi-isometric to either the mapping class group equipped with the word metric, or to Teichmuller space equipped with either the Teichmuller metric or the Weil-Petersson metric. We introduce a unified approach to study the coarse geometry of these spaces. We show that the quasi-Lipschitz image in X of a box in R^n is locally near a standard model of a flat in X. As a consequence, we show that, for all these spaces, the geometric rank and the topological rank are equal. The methods are axiomatic and apply to a larger class of metric spaces.

Some corrections have been made. Also, the coarse differentiation statement has been modified to state that a quasi-Lipschitz map is "differentiable almost everywhere"

References in corpus (4)

• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Uniform hyperbolicity of the curve graph via surgery sequences
• Large scale rank of Teichmuller space
• The augmented marking complex of a surface

Cited by in corpus (10)

• Hierarchically hyperbolic spaces I: curve complexes for cubical groups
• Boundaries and automorphisms of hierarchically hyperbolic spaces
• Large scale rank of Teichmuller space
• Geodesics in the mapping class group
• Stable cubulations, bicombings, and barycenters
• The asymptotic geometry of the Teichmüller metric: Dimension and rank
• Sublinearly Morse Boundary II: Proper geodesic spaces
• Rigidity of Teichmüller space
• Extensions of multicurve stabilizers are hierarchically hyperbolic
• Asymptotic Dimension of Big Mapping Class Groups