AI 中文总结
研究不可定向曲面裤子图与其可定向双覆盖裤子图的关系，通过提升裤子分解构造映射，证明其为拟等距嵌入，进而得出不可定向曲面裤子图与配备诱导度量的泰希米勒空间拟等距。
AI 中文摘要
我们研究不可定向曲面的裤子图与其可定向双覆盖的裤子图之间的关系。给定具有可定向双覆盖\pi:S→N的不可定向曲面N，我们通过提升裤子分解构造裤子图之间的自然映射。证明该映射定义了Pants(N)到Pants(S)的拟等距嵌入。利用Brock关于Pants(S)与配备韦伊 - 彼得森度量的泰希米勒空间Teich(S)之间的拟等距，以及N的泰希米勒空间与Teich(S)上甲板对合的不动点轨迹的等同，证明Pants(N)与配备诱导韦伊 - 彼得森度量的N的泰希米勒空间拟等距。
英文摘要
We study the relation between the pants graph of a non-orientable surface and that of its orientable double cover. Given a non-orientable surface $N$ with orientable double cover $π\colon S\to N$, we construct a natural map between pants graphs induced by lifting pants decompositions. We prove that this map defines a quasi-isometric embedding of $\mathrm{Pants}(N)$ into $\mathrm{Pants}(S)$. Using Brock's quasi-isometry between $\mathrm{Pants}(S)$ and Teichmüller space $\mathrm{Teich}(S)$ endowed with the Weil-Petersson metric, together with the identification of the Teichmüller space of $N$ with the fixed-point locus of the deck involution on $\mathrm{Teich}(S)$, we prove that $\mathrm{Pants}(N)$ is quasi-isometric to the Teichmüller space of $N$ equipped with the induced Weil-Petersson metric.
