Finding applications of dilogarithm identities on hyperbolic surfaces
Speaker: Đoàn Nhật Minh

Time: 14h00, Thursday, November 11, 2021 (Vietnam time).

Link Online: meet.google.com/zsh-jnxc-eit

Abstract: I will talk about an investigation on the terms arising in the Luo-Tan 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. 

If time allows, the second part of the talk will be about my current project. In particular, I will describe a tree structure of the set of orthogeodesics on hyperbolic surface. Using this tree structure, we are able to give a combinatorial proof of the Basmajian identity. As another application, dilogarithm identities following from the Bridgeman identity are computed recursively and their terms are indexed by Farey sequence. We also introduce the notion of r-orthoshapes with associated identity relations and indicate connections to length equivalent orthogeodesics and a Diophantine equation.