用Farey符号计算$PSL_2(\mathbb Z)$的有限指数子群
Computations with finite index subgroups of $PSL_2(\mathbb Z)$ using Farey Symbols
摘要：
模群的有限指数子群具有重要的算术意义。1991年由Ravi Kulkarni引入的Farey符号是与这些群体合作的工具。给定这样的群$Γ$，$Γ$的Farey符号是有理数的某个有限序列（表示$Γ$的基本区的顶点）以及顶点之间的边的配对信息。它们是一种紧凑的编码群信息的方式，并且提供了一种简单的计算群的方法。举例来说：计算生成元的独立集，并将群元素分解为这些生成元中的一个字，找到陪集代表、椭圆点和群的亏格，测试群是否是同余等。在这篇临时文章中，我们将讨论Farey符号和使用它们的显式算法。
英文摘要：
Finite index subgroups of the modular group are of great arithmetic importance. Farey symbols, introduced by Ravi Kulkarni in 1991, are a tool for working with these groups. Given such a group $Γ$, a Farey symbol for $Γ$ is a certain finite sequence of rational numbers (representing vertices of a fundamental domain of $Γ$) together with pairing information for the edges between the vertices. They are a compact way of encoding the information about the group and they provide a simple way to do calculations with the group. For example: calculating an independent set of generators and decomposing group elements into a word in these generators, finding coset representatives, elliptic points, and genus of the group, testing if the group is congruence, etc. In this expository article, we will discuss Farey Symbols and explicit algorithms for working with them.
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