Wednesday, April 24, 2024 2:30pm to 3:20pm
About this Event
Speaker: Dragomir Saric (Graduate Center of CUNY and Queens college)
Abstract: A hyperbolic geodesic in the unit disk model of the hyperbolic plane is uniquely determined by the pair of its ideal endpoints on the unit circle. A circle homeomorphism maps the pairs of endpoints of the sides of the Farey triangulation onto arbitrary pairs of points on the unit circle. Thus, a circle homeomorphism extends to a map from the Farey triangulation to another triangulation of the unit disk. A configuration of two adjacent ideal hyperbolic triangles (up to isometry) is determined by a unique real number called the shear. Penner's idea was to parametrize various smoothness classes of circle homeomorphisms using various conditions on the shears of the edges. We describe our results that characterize shear functions that give rise to homeomorphisms and quasisymmetric and symmetric maps of the circle. In a joint work with H. Parlier, we describe the connection between the shear coordinates, infinite diagonal flips on the Farey triangulation, and a sufficient condition for quasisymmetry in terms of shears given by Penner and Sullivan. In a joint work with C. Wolfram and Y. Wang, we give a sufficient condition on the shears such that the induced homeomorphism is in the Weil-Petersson class.
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