论文标题
外层空间的热力学指标
Thermodynamic metrics on outer space
论文作者
论文摘要
在本文中，我们考虑了在Culler-Vogtmann外层空间上定义的两个分段Riemannian指标，我们称之为熵指标和压力指标。由于McMullen的工作，这些指标可以看作是封闭表面Teichmüller空间上Weil-Petersson指标的类似物。我们表明，尽管从几何群体理论的角度来看，这些指标的几何分析与Weil-Petersson指标的几何分析相似，但这些指标的行为与Weil-Petersson指标截然不同。具体来说，我们表明，当$ r $至少4级时，使用熵公制的Culler-Vogtmann外部空间完成$ {\ rm out}（\ mathbb {f} _r）$的操作（\ Mathbb {f} _r）$具有固定点。压力度量也有类似的陈述。
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.
