Symmetries of Riemann surfaces with large automorphism group
Symmetries of Riemann surfaces with large automorphism group
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具有大自同构群的黎曼曲面的对称性
DOI：
10.1007/bf01344543
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发表时间：
1974
影响因子：
1.4
通讯作者：
D. Singerman
中科院分区：
文献类型：
--
作者：
D. Singerman
一个黎曼曲面是对称的，如果它允许一个反共形对合。本文讨论的基本问题是亏格为g> t的紧致Riemann曲面是否是对称的，其中亏格为g> t的Riemann曲面上的自同构群为大群。已知亏格g> 1的紧致Riemann曲面的自同构群是有限的，且上界为84（g-1）。Macbeath（[t21 13]）已经找到了无穷多个g，对于这些g可以达到这个界。我们发现，所有的表面发现Macbeath的方法确实是对称的。然而，我们展示了一个例子的非对称黎曼曲面的属g=！7的Riemann曲面，该曲面允许84（g-1）个自同构，我们还研究了允许高阶自同构的Riemann曲面。亏格为g的黎曼曲面的自同构的阶以4g+ 2为上界，并且对每个g都达到这个界[8]。我们证明了，所有的黎曼曲面承认自同构的阶大于2g+ 2是对称的。我们的工作与曲面上的非自反正则映射理论有着密切的联系。(See（8）定义。正则映射群与紧致黎曼曲面的自同构群之间存在联系。事实上，亏格g大于24（g-1）的黎曼曲面的每个自同构群也是某个正则映射的群，反之，正则映射的每个群可以被认为是黎曼曲面的自同构群。无自反的正则映射是相当特殊的。(In事实上，在[3]的早期版本中提出，对于亏格O> 1）的曲面，它们不存在。我们在上面的对应中表明，非对称曲面的自同构的大群将产生不可自反的正则映射，但这一事实的匡威并不总是正确的。例如，亏格为g的紧致非对称黎曼曲面的阶数大于24（g-1）的自同构群比非自反正则映射更例外。对称黎曼曲面还有另一种有趣的解释。每一个紧致黎曼曲面都可以作为代数曲线f（z，w）= 0的黎曼曲面得到。黎曼曲面
A Riemann surface is symmetric if it admits an anti-conformal involution. The basic question which we discuss in this paper is whether compact Riemann surfaces of genus g> t which admit large groups of automorphisms are symmetric. As is weU-known, the automorphism group of a compact Riemann surface of genus g> 1 is finite and bounded above by 84 (g-1). Macbeath ([t21 13]) has found infinitely many g for which this bound is attained. We show that all the surfaces found by Macbeath's methods are indeed symmetric. However, we do exhibit an example of a non-symmetric Riemann surface of genus g=! 7 which does admit 84 (g-1) automorphisms.We also study Riemann surfaces admitting automorphisms of large order. The order of an automorphism of a Riemann surface of genus g is bounded above by 4g+ 2 and this bound is attained for every g [8]. We show that all Riemann surfaces admitting automorphisms of order greater that 2g+ 2 are symmetric. There is a close link between our work and the theory of irreflexible regular maps on surfaces.(See § 8 for definitions.) There is a connection between the groups of regular maps and large groups of automorphisms of compact Riemann surfaces. Indeed, every group of automorphisms ofa Riemann surface of genus g of order greater than 24 (g-1) is also the group of some regular map and conversely, every group of a regular map can be thought of as the group of automorphisms of a Riemann surface. The irreflexible regular maps turn out to be rather exceptional.(In fact, it was suggested in early editions of [3] that they did not exist for surfaces of genus O> 1). We show in the above correspondence that large groups of automorphisms of non-symmetric surfaces will give rise to irreflexible regular maps, but that the converse of this fact is not always true. Thus, for example, groups of automorphisms of order greater than 24 (g-1) of a compact non-symmetric Riemann surface of genus g are more exceptional than irreflexible regular maps. There is another interpretation of symmetric Riemann surfaces which is of interest. Every compact Riemann surface can be obtained as the Riemann surface of an algebraic curve f (z, w)= 0. A Riemann surface