Pressure type metrics on spaces of metric graphs

TL;DR

This paper compares two Riemannian metrics on a moduli space of metric graphs to the classical Weil-Petersson metric, highlighting their geometric features and differences.

Contribution

It introduces and analyzes two new metrics on the moduli space of metric graphs as analogues of the Weil-Petersson metric, expanding understanding of geometric structures in this setting.

Findings

Comparison of geometric features of the two metrics

Identification of similarities with Weil-Petersson metric

Insights into the structure of the moduli space of metric graphs

Abstract

In this note, we consider two Riemannian metrics on a moduli space of metric graphs. Each of them could be thought of as an analogue of the Weil-Petersson metric on the moduli space of metric graphs. We discuss and compare geometric features of these two metrics with the "classic" Weil-Petersson metric in Teichm\"uller theory. This paper is motivated by Pollicott and Sharp's earlier work.