Weil-Petersson analogs for graphs

The Weil-Petersson metric is a Riemannian metric on the Teichmuller space which comes from and reflects the geometry of the hyperbolic metrics on the underlying surface. Motivated by foundational work of McMullen, Pollicott-Sharp (and later Kao) proposed an analogous metric for the moduli space of metrics on a fixed graph. We study this metric and completely characterize its completion in the case of a rose, showing that it resembles the completion of the classical WP metric. This represents joint work with Matt Clay and Yo’av Rieck.