论文标题
计数双曲多地点相对于单个组件的长度
Counting hyperbolic multi-geodesics with respect to the lengths of individual components
论文作者
论文摘要
鉴于有连接的，定向的，完整的，有限面积的超质表面$ x $ x $属$ g $带有$ n $ punctures，米尔扎卡尼（Mirzakhani $ 6G-6+2N $。我们在映射简单或填充封闭的多曲线的类轨道中，建立同类的渐近学，以计算多地点的计算，这些轨道可以跟踪单个成分的双曲线长度，从而证明和推广沃尔珀特的猜想。在简单的情况下，我们考虑了更精确的计数，这些计数也可以跟踪投影测量的测量层压板的多地点类别。我们提供了所有所考虑的计数的渐近术语的统一几何和拓扑描述。我们的证明结合了米尔扎卡尼（Mirzakhani）的几篇论文以及马古利斯（Margulis）在他的论文中提出的思想的结果。
Given a connected, oriented, complete, finite area hyperbolic surface $X$ of genus $g$ with $n$ punctures, Mirzakhani showed that the number of multi-geodesics on $X$ of total hyperbolic length $\leq L$ in the mapping class group orbit of a given simple or filling closed multi-curve is asymptotic as $L \to \infty$ to a polynomial in $L$ of degree $6g-6+2n$. We establish asymptotics of the same kind for countings of multi-geodesics in mapping class group orbits of simple or filling closed multi-curves that keep track of the hyperbolic lengths of individual components, proving and generalizing a conjecture of Wolpert. In the simple case we consider more precise countings that also keep track of the class of the multi-geodesics in the space of projective measured geodesic laminations. We provide a unified geometric and topological description of the leading terms of the asymptotics of all the countings considered. Our proofs combine techniques and results from several papers of Mirzakhani as well as ideas introduced by Margulis in his thesis.
