Theory of Maps on Orientable Surfaces
Theory of Maps on Orientable Surfaces
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DOI：
10.1112/plms/s3-37.2.273
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发表时间：
1978-09
影响因子：
1.8
通讯作者：
G. Jones;D. Singerman
中科院分区：
文献类型：
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作者：
G. Jones;D. Singerman
我们使用的想法（由于Heffter和最近归因于埃德蒙兹）相关联的地图dt上的定向表面SP的一对排列x，y的集合Q。飞镖是反转每个箭头方向的对合，y通过遵循围绕v的方向循环地置换指向每个顶点v的箭头。这个想法被广泛地用于证明Heawood地图着色定理（例如，见[19]），重点放在排列x和y上，而不是放在它们生成的群G上;在我们的例子中，正如Biggs [2]的工作一样，0及其对O的作用。是根本性的。存在一个明显的满射6：F-+ 0，其中F是三角形群（2，m，n），对于某个m，n（可能是无限的），表示为gp< X，Y\X2= Ym=（F^ X）*= 1>.现在F可以表示为一组单连通黎曼曲面的共形变换，留下不变的三角形镶嵌。如果我们定义JV的映射子群为M= 6-x（Ga），其中Ga是某个EQ的G中的稳定子，则商曲面ti/M同胚于 *^的基础曲面ε f，并且通过使用tt的镶嵌，我们可以构造关于π/M的映射M，该映射同构于Jt。首先，群（2，m，n）的映射与Schreier陪集图之间存在着自然的对应关系。这暗示了通过考虑除了（2，m，n）形式的群之外的其他群类的一些明显的推广。事实上，这些有时会产生已经考虑过的组合对象。为
We use the idea (due to Heffter and more recently attributed to Edmonds) of associating with a map dt on an orientable surface SP a pair of permutations x, y of the set Q. of darts (directed edges) of Jt: a; is the involution which reverses the direction of each dart and y cyclically permutes the darts directed towards each vertex v by following the orientation around v. This idea was widely used in proving the Heawood map-colouring theorem (see [19] for instance) with the emphasis on the permutations x and y rather than on the group G they generate; in our case, as in the work of Biggs [2], 0 and its action on O. are fundamental. There is an obvious epimorphism 6: F-+ 0 where F is the triangle group (2, m, n) with presentation gp< X, Y\X2= Ym=(F^ X)*= 1> for some m, n (which may be infinite). Now F can be represented as a group of conformal transformations of a simply-connected Riemann surface<^, leaving invariant a triangular tessellation of<%. If we define a mapsubgroup for JV to be M= 6-x (Ga), where Ga is the stabilizer in G of some a E Q., then the quotient surface ti/M is homeomorphic to the underlying surface£ f of*^, and by using the tessellation of tt we can construct a map M on%/M isomorphic to Jt.This line of thought has the following consequences. Firstly, there is a natural correspondence between maps and Schreier coset-graphs for the groups (2, m, n). This suggests some obvious generalizations by considering other classes of groups than just those of the form (2, m, n). Indeed, these sometimes produce combinatorial objects already considered. For