A Weil-petersson Type Metric on Spaces of Metric Graphs

Given a compact topological surface V with negative Euler characteristic, the Teichmüller space Teich(V ) describes the marked Riemann metrics (with constant curvature κ = −1) which it supports. More precisely, Teich(V ) is the set of equivalence classes (Vg, φ), where Vg is a hyperbolic surface with Riemann metric g and φ : V → Vg is a homeomorphism, with (Vg1 , φ1) equivalent to (Vg2 , φ2) if there is an isometry ψ : Vg1 → Vg2 such that ψ ◦ φ1 is isotopic to φ2. The moduli space Mod(V ) describes the unmarked Riemann metrics on V and is obtained by quotienting Teich(V ) by the Mapping Class Group of V . There are several different metrics which can naturally be defined on Teich(V ), for example, the Teichmüller metric and the Weil-Petersson metric, both of which are invariant under the Mapping Class Group and descend to Mod(V ). There is a particularly illuminating formulation of the Weil-Petersson metric, due to Wolpert, in terms of the second derivative of the length of a typical geodesic on V [Wo]. Recently, a more dynamical characterization of this was proposed by Curt McMullen, who thereby extended the notion of the Weil-Petersson metric to a variety of settings (e.g., from Fuchsian to QuasiFuchsian groups) [Mc]. In this note we will introduce an analogue of the Weil-Petersson metric for families of metric graphs, and explore its properties through some simple examples. To formulate an analogous definition for families of metric graphs we can replace the surface V by a finite (undirected) graph G with edge set E . We can replace the Riemann metrics by edge weightings l : E → R.