Bowditch and Epstein described a flow on the Teichmuller space, whose large time limit describes hyperbolic surfaces with large geodesic boundaries in terms of their spine. The spine is an isotopy class of metric graphs embedded in the surface and onto which the surface retracts. The space of such structures is called the combinatorial Teichmuller space, and it is equipped with a symplectic form used by Kontsevich in his proof of the Witten's conjecture. Upgrading results of Mondello, Luo, Do, we show that many aspects of hyperbolic geometry have an analog in the combinatorial geometry and that the latter are the large time limits of the former under the flow: Fenchel-Nielsen coordinates, Wolpert's formula for the symplectic form, 9g - 9 + 3n theorem, Mirzakhani-McShane identities, .... As
corollaries we obtain (1) a piecewise-linear structure on the combinatorial Teichmuller space, and (2) geometric proofs of topological recursion for the symplectic volumes (Kontsevich) and the lattice point count (Norbury) of the combinatorial Teichmuller space, which are parallel to Mirzakhani's proof of the topological recursion for the Weil-Petersson volumes.
This is joint work with Jorgen Ellegaard Andersen, Severin Charbonnier, Alessandro Giacchetto, Danilo Lewanski, Campbell Wheeler.