Title: On a tree structure of the set of orthogeodesics on hyperbolic surfaces

Time: 14:00 (VN), Friday October 29, 2021

Online at https://meet.google.com/fkz-fbuf-cvz

Abstract: The set of orthogeodesics, introduced by Basmajian in the early 90's, is the set of geodesic arcs perpendicular to the boundary of a hyperbolic surface at their ends. Basmajian's and Bridgeman's identities are two identities connecting the ortholength spectrum with the total length of boundary and the area of a hyperbolic surface. We will describe a tree structure on the set of orthogeodesics and give a combinatorial proof of Basmajian's identity. As another application, (dilogarithm) identities following from Basmajian's identity and Bridgeman's 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 Penner's Ptolemy relation, length equivalent orthogeodesics and a Diophantine equation.