Conjugators of Fuchsian groups and quasiplatonic surfaces
Conjugators of Fuchsian groups and quasiplatonic surfaces
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Fuchsian 群和准柏拉图曲面的共轭子
DOI：
10.1093/qmath/hah054
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发表时间：
2005
影响因子：
0.7
通讯作者：
J. Wolfart
中科院分区：
文献类型：
--
作者：
E. Girondo;J. Wolfart
摘要设G是一个包含两个定义同构黎曼曲面的无扭转子群的Fuchsian群。那么这些表面子群K和αKα−1在psl （2， R）中是共轭的，但一般来说共轭元α不能在G或G的有限指数傅氏扩展中取。我们将证明，在三角群G中的正规包含的情况下，这些α可以在某个扩展G的三角群中选择。结果表明，导致这一结果的方法也允许回答在给定的拟柏拉图黎曼曲面上可以存在多少种相同类型的不同正则波的问题。g > 1属的拟柏拉图黎曼曲面，其表面群K是三角形群∆的正规子群。众所周知，对于每一个属，只有有限多个拟柏拉图曲面的同构类，并且当且仅当它们的曲面群K，K 0在psl （2， R）中共轭时，两个拟柏拉图曲面是同构的。因此，所有关于准柏拉图曲面族上的分类和伽罗瓦作用的问题都会导致以下类型的问题
AbstractLet G be a Fuchsian group containing two torsion free subgroups defining isomor-phic Riemann surfaces. Then these surface subgroups K and αKα − 1 are conjugate inPSL(2 , R), but in general the conjugating element α cannot be taken in G or a finite indexFuchsian extension of G . We will show that in the case of a normal inclusion in a trianglegroup G these α can be chosen in some triangle group extending G . It turns out thatthe method leading to this result allows also to answer the question how many differentregular dessins of the same type can exist on a given quasiplatonic Riemann surface. 1 Introduction Quasiplatonic Riemann surfaces of genus g > 1 can be characterized by the fact that theirsurface groups K are normal subgroups of triangle groups ∆. It is well known that for everygenus there are only finitely many isomorphism classes of quasiplatonic surfaces, and thattwo such surfaces are isomorphic if and only if their surface groups K,K 0 are conjugate inPSL(2 , R). All questions concerning classification and Galois actions on families of quasi-platonic surfaces lead therefore to problems of the following type: Let