Currents with corners and counting weighted triangulations

Abstract

Let $\Sigma$ be a closed orientable hyperbolic surface. We introduce the notion of a \textit{geodesic current with corners} on $\Sigma$, 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. 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 $\Sigma$. This leads to the space of \textit{marked harmonic currents with corners}, which admits a natural Borel action by the mapping class group, and an analog of Bonahon's\cite{Bonahon} compactness criterion for sub-level sets of the intersection form against a filling current. As an application, we consider an analog of a curve counting problem on $\Sigma$ for triangulations. Fixing an embedding $\phi$ of a weighted graph $\Gamma$ into $\Sigma$ whose image $\phi(\Gamma)$ is a triangulation of $\Sigma$, let $N_{\phi}(L)$ denote the number of mapping classes $f$ so that a weighted-length minimizing representative in the homotopy class determined by $f \circ \phi$ has length at most $L$. In analogy with theorems of Mirzakhani\cite{Mirzakhani}, Erlandsson-Souto\cite{ErlandssonSouto}, and Rafi-Souto\cite{RafiSouto}, we prove that $N_{\phi}(L)$ grows polynomially of degree $6g-6$ and the limit \[ \lim_{L \rightarrow \infty} \frac{N_{\phi}(L)}{L^{6g-6}}\] exists and has an explicit interpretation depending on the geometry of $\Sigma$, the vector of weights, and the combinatorics of $\phi$ and $\Gamma$.