Quantifying some properties of curves and arcs on hyperbolic surfaces

Motivated by the ergodicity of geodesic flow on the unit tangent bundle of a closed hyperbolic surface and its applications, this thesis includes three parts:
Part 1. We present a type of quantitative density of closed geodesics and orthogeodesics on complete finite-area hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic and the shortest doubly truncated orthogeodesic that are Y-dense on a given compact set on the surface.
Part 2. We investigate the terms arising in Luo-Tan’s identity, namely showing that they vary monotonically in terms of lengths and that they verify certain convexity properties. Using these properties, we deduce two results. As a first application, we show how to deduce a theorem of Thurston which states, in particular for closed hyperbolic surfaces, that if a simple length spectrum "dominates" another, then in fact the two surfaces are isometric. As a second application, we show how to find upper bounds on the number of pairs of pants of bounded length that only depend on the boundary length and the topology of the surface. This is joint work with Hugo Parlier and Ser Peow Tan.
Part 3. Inspired by a number theoretic application of Bridgeman’s identity, the combinatorial proof of McShane’s identity by Bowditch and its generalized version by Labourie and Tan, we describe a tree structure on the set of orthogeodesics and give a combinatorial proof of Basmajian’s identity in the case of surfaces. We also introduce the notion of orthoshapes with associated identity relations and indicate connections to length equivalent orthogeodesics and a type of Cayley-Menger determinant. As another application, dilogarithm identities following from Bridgeman’s identity are computed recursively and their terms are indexed by the Farey sequence.