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 Teichmüller space of a closed surf...

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Bibliographic Details
Published in:Ergodic theory and dynamical systems 2023-03, Vol.43 (3), p.729-793
Main Authors: AOUGAB, TARIK, CLAY, MATT, RIECK, YO’AV
Format: Article
Language:English
Subjects:
Entropy
Group theory
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Summary: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 than the Weil–Petersson metric. Specifically, we show that when the rank r is at least 4, the action of $\operatorname {\mathrm {Out}}(\mathbb {F}_r)$ 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.
ISSN:0143-3857
1469-4417
DOI:10.1017/etds.2021.165