Thermodynamic metrics on outer space

Tarik Aougab,Matt Clay,Yo’Av Rieck

doi:10.1017/etds.2021.165

Abstract

AbstractIn 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.

Highlights

The purpose of this paper is to introduce and examine two piecewise Riemannian metrics, called the entropy metric and the pressure metric, on the Culler–Vogtmann outer space CV (Fr )
The Culler–Vogtmann outer space is the moduli space of unit-volume marked metric graphs and as such it is often viewed as the analog of the Teichmüller space of an orientable surface Sg
As the piecewise Riemannian metrics on the Culler–Vogtmann outer space that we study in this paper are motivated by the classical Weil–Petersson metric on the Teichmüller space of a closed surface, it is natural to ask to what extent they are true analogs of the Weil–Petersson metric

Summary

Introduction

The purpose of this paper is to introduce and examine two piecewise Riemannian metrics, called the entropy metric and the pressure metric, on the Culler–Vogtmann outer space CV (Fr ). That unit-entropy metrics on subroses arise as points in the completion follows from the calculations provided for the proof of incompleteness in Theorem 1.1 and some continuity arguments One can consider an entropy function defined over the moduli space of singular flat metrics on a closed surface This setting appears more similar to the situation of Theorem 1.3 in that the unit-entropy condition is not encoded by the local geometry. Positive definiteness of the Hessians follows from strict convexity of hG and PG on M(G) and R|E+|, respectively (Theorem 3.7) Using these norms, we can define the entropy or pressure length of a piecewise smooth path t : [t0, t1] → M1(G) by t1. As the closure of this set in X1(F2) is compact, the sequence (xn)n∈N converges

The moduli space of the rose

10. The moduli space of a graph with a separating edge