Farey词及其迹的组合数学
The combinatorics of Farey words and their traces
AI总结：
本文引入与3-流形和orbifold几何拓扑相关的三元Farey多项式族，通过抽象组合观点研究其递归定义和性质，为PSL(2,C)中秩二子群的分类问题提供实用方法。
摘要：
我们引入了一个与$3$-流形和orbifold的几何与拓扑密切相关的三元“Farey多项式”族，因为它们可用于生成Kleinian群的一维复形变空间的边界和局部坐标的具体实现。因此，这个多项式族具有许多相当显著的性质。我们从抽象组合的角度研究这些多项式，包括扩展文献中已知的流形特例的递归定义，甚至超越了几何预测的范围。我们还提出了一些有趣的例子和猜想，希望引起对代数组合学和超几何函数感兴趣的研究人员的注意。本文的结果还为PSL(2,C)中秩二子群的各种分类问题提供了实用方法，因为结合作者的其他近期工作，这些结果能够为某些群是离散且自由的提供证明，并提供识别关系子的有效方法。
英文摘要：
We introduce a family of 3-variable "Farey polynomials" that are closely connected with the geometry and topology of $3$-manifolds and orbifolds as they can be used to produce concrete realisations of the boundaries and local coordinates for one-complex-dimensional deformation spaces of Kleinian groups. As such, this family of polynomials has a number of quite remarkable properties. We study these polynomials from an abstract combinatorial viewpoint, including a recursive definition extending that which is known in the literature for the special case of manifolds, even beyond what the geometry predicts. We also present some intriguing examples and conjectures which we would like to bring to the attention of researchers interested in algebraic combinatorics and hypergeometric functions. The results in this paper additionally provide a practical approach to various classification problems for rank-two subgroups of PSL(2,C) since they, together with other recent work of the authors, make it possible to provide certificates that certain groups are discrete and free, and effective ways to identify relators.
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