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arXiv · 2009.13314

Thermodynamic metrics on outer space

Tarik Aougab·Matt Clay·Yo'av Rieck
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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 Teichmüller 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.

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Tarik Aougab, Matt Clay, Yo'av Rieck. 2020-09-28. Thermodynamic metrics on outer space. https://arxiv.org/abs/2009.13314

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