Farey triangulation, shears and circle maps

Dragomir Saric (CUNY)

Abstract: We define shears as a discrete geometric/analytic quantity that describes how circle homeomorphisms change the standard Farey triangulation of the disk into a non-standard triangulation. Conversely, given an assignment of shears on the Farey triangulation a natural question posed by Penner and first considered by Penner and Sullivan is to find conditions on the shears that guarantee the induced map of the circle is a homeomorphism, or a quasisymmetric map or some other smoothness class of interest. We first describe our results on characterizing homeomorphisms, quasisymmetric and symmetric maps of the circle in terms of shears on the Farey triangulation. In recent joint study with Hugo Parlier, we consider an infinite diagonal flips on the Farey triangulation as a particular method of deforming the Farey triangulation. This is motivated by the study of the big mapping class group. We find that the induced circle homeomorphisms are quasisymmetric and to our surprise, they belong to a subclass of quasisymmetric maps first introduced by Penner and Sullivan. This subclass was introduced by the requirement that the image triangulation has a choice of horocycles at each vertex that together with the edges of the triangulation form bounded hexagons in the unit disk. We find a direct and simple characterization of this class only in terms of shears (without the need to construct horocycles). If time permits, I will also discuss recent work with Yilin Wang and Catherine Wolfram on the relationship of the Weil-Petersson homeomorphism class and diamond shears on the Farey triangulation.