Large scale rank of Teichmuller space

TL;DR

This paper introduces a unified approach to studying the coarse geometry of Teichmuller space and related spaces, showing that their geometric and topological ranks are equal, with methods applicable to broader metric spaces.

Contribution

It provides a new axiomatic framework to analyze the coarse geometry of Teichmuller space and similar spaces, establishing the equality of geometric and topological ranks.

Findings

Quasi-Lipschitz images of R^n boxes are near standard flats.

Geometric and topological ranks are equal for these spaces.

Methods are applicable to a broader class of metric spaces.

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.