Thermodynamic metrics on outer space

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