Thermodynamic metrics on outer space

In this paper we consider two piecewise Riemannian metrics defined on the Culler-Vogtmann outer space which we call the entropy metric and the pressure metric. As a result of work of McMullen, these metrics can be seen as analogs of the Weil-Petersson metric on the Teichm\"uller space of a closed surface. We show that while the geometric analysis of these metrics is similar to that of the Weil-Petersson metric, from the point of view of geometric group theory, these metrics behave very differently to the Weil-Petersson metric. Specifically, we show that when the rank $r$ is at least 4, the action of ${\rm Out}(\mathbb{F}_r)$ on the completion of the Culler-Vogtmann outer space using the entropy metric has a fixed point. A similar statement also holds for the pressure metric.