Existence and Non-Existence of Torsion in Maximal Arithmetic Fuchsian Groups
Existence and Non-Existence of Torsion in Maximal Arithmetic Fuchsian Groups
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极大算术Fuchsian群中挠率的存在与不存在
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发表时间：
2009
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通讯作者：
C. Maclachlan
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作者：
C. Maclachlan
在[1]中，Borel 讨论了由数域上的四元数代数产生的离散算术群，特别是算术 Kleinian 群和算术 Fuchsian 群。在这些情况下，他在每个可通约性类中描述了包含所有最大群的群类。 Chinburg 和 Friedman [2] 开发了将定义数域的交换阶嵌入定义四元数代数中的最大或艾希勒阶的结果，并根据定义算术数据陈述了此类群中存在挠率的充分必要条件。这在 [3] 中的克莱因群的情况下得到了更充分的探讨。在 Fuchsian 群的情况下，这些关于挠率存在性的结果被扩展以获得该类中每个群的有限循环子群的共轭类数的公式 [8, 9]。在本文中，我们在算术 Fuchsian 群的范围内研究了最大 Fuchsian 群中挠率的分布范围。一些低属案例的研究（参见例如[7, 12]）表明2-扭转非常普遍。这里获得的结果证实了这一点，但我们还将获得无挠的最大算术 Fuchsian 群。作者感谢 Alan Reid 就本文部分内容进行的对话。
In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.