Thermodynamic metrics on outer space

TL;DR

This paper explores two new metrics on outer space, revealing their geometric similarities to known metrics but also highlighting distinct behaviors in group actions, especially for ranks at least 4.

Contribution

It introduces the entropy and pressure metrics on outer space and analyzes their geometric and group-theoretic properties, contrasting them with the Weil-Petersson metric.

Findings

The entropy and pressure metrics are analogous to the Weil-Petersson metric.

For rank r ≥ 4, the Out(F_r) action on the metric completion has a fixed point.

These metrics behave differently from Weil-Petersson in geometric group theory contexts.

Abstract

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.