Quasisymmetric maps, shears, lambda lengths and flips

TL;DR

This paper characterizes quasisymmetric maps on the hyperbolic plane boundary through their effects on Farey triangulation, lambda lengths, and flip distances, extending prior work by Penner and Sullivan.

Contribution

It precisely identifies which quasisymmetric maps correspond to pinched lambda lengths using shearing coordinates and flip distances, clarifying earlier results.

Findings

Characterization of quasisymmetric maps via shearing coordinates

Identification of maps with pinched lambda lengths

Correlation between flip distance and quasisymmetric maps

Abstract

We study quasisymmetric maps, which act on the boundary of the hyperbolic plane, by looking at their action on the Farey triangulation. Our main results identify exactly which quasisymmetric maps correspond to pinched lambda lengths in terms of shearing coordinates and, separately, in terms of (simultaneous) flip distance of the Farey triangulation and its image by the map. These extend and clarify previous results of Penner and Sullivan.