Geometry and Topology Seminar (1/2)

We will describe some curious stability phenomena that appear when comparing lengths of curves on different hyperbolic surfaces. In particular, we will explain construct "noisy" families of geodesics for Thurston's Lipschitz metric on Teichmüller space. We then explain a construction of n-ples of Fuchsian representations where an invariant built out of vectors of lengths of curves, called the limit cone, exhibits a surprising finite-sidedness behavior. If time allows, we will conclude by discussing the estimates on lengths of curves that we use to prove polyhedrality of our examples of limit cones.