Currents with corners and counting weighted triangulations

Abstract Let be a closed orientable hyperbolic surface. We introduce the notion of a geodesic current with corners on , which behaves like a geodesic current away from certain singularities (the “corners”). We topologize the space of all currents with corners and study its properties. In several ways that we make precise, the space of geodesic currents is “at infinity” in the space of currents with corners, in the sense that they arise as limits in which the corners are pushed out to infinity. We also emphasize the following analogy: currents are to closed curves, as currents with corners are to graphs on . We prove that the space of currents with corners shares many properties with the space of geodesic currents, although crucially, there is no canonical action of the mapping class group nor is there a continuous intersection form. To circumvent these difficulties, we focus on those currents with corners arising from harmonic maps of graphs into . This leads to the space of marked harmonic currents with corners , which admits a natural Borel action by the mapping class group, and an analog of Bonahon's compactness criterion for sublevel sets of the intersection form against a filling current. As an application, we consider an analog of a curve counting problem on for triangulations. Fixing an embedding of a weighted graph into whose image is a triangulation of , let denote the number of mapping classes so that a weighted‐length minimizing representative in the homotopy class determined by has length at most . In analogy with theorems of Mirzakhani, Erlandsson–Souto, and Rafi–Souto, we prove that grows polynomially of degree and the limit exists and has an explicit interpretation depending on the geometry of , the vector of weights, and the combinatorics of and .