外空间上的完全不变测地线度量与热带曲线的雅可比簇
Complete invariant geodesic metrics on outer spaces and Jacobian varieties of tropical curves
摘要：
设$\mathrm{Out}(F_n)$是自由群$F_n$的外自同构群。它适当地作用在标记度量图的外层空间$X_n$上，这是一个有限维的无限单纯复形，其中缺少一些单纯面。本文构造了$X_n$上的完备测地线度量和完备分片光滑黎曼度量，它们在$\mathrm{Out}(F_n)$下是不变的.一个关键的组成部分是识别度量图与热带曲线和使用的热带雅可比映射从模空间的热带曲线的模空间的主要极化热带阿贝尔簇。
英文摘要：
Let $\mathrm{Out}(F_n)$ be the outer automorphism group of the free group $F_n$. It acts properly on the outer space $X_n$ of marked metric graphs, which is a finite-dimensional infinite simplicial complex with some simplicial faces missing. In this paper, we construct complete geodesic metrics and complete piecewise smooth Riemannian metrics on $X_n$ which are invariant under $\mathrm{Out}(F_n)$. One key ingredient is the identification of metric graphs with tropical curves and the use of the tropical Jacobian map from the moduli space of tropical curves to the moduli space of principally polarized tropical abelian varieties.
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