Calculation of Fuchsian groups associated to billiards in a rational triangle
Calculation of Fuchsian groups associated to billiards in a rational triangle
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有理三角形中与台球相关的 Fuchsian 群的计算
DOI：
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发表时间：
1998
影响因子：
0.9
通讯作者：
C. Ward
中科院分区：
文献类型：
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作者：
C. Ward
我们定义，以下Veech，Fuchsian组$伽玛（P）$的一个合理的多边形$P$。如果$P$是单连通的，那么“有理”等价于$P$的所有内角都是$pi$的有理倍数的条件。如果$Gamma（P）$在$mathop{
m PSL}
olimits（2，{Bbb R}）$（因此是一个{it格}），那么维奇的一个定理指出，$P$中的每一条台球路径要么是有限的，要么是均匀分布的。我们考虑各种有理三角形的Fuchsian群。首先，我们明确地计算了一个新的三角形序列的Fuchsian群，并发现它们是格。有趣的是，所发现的晶格与先前已知的晶格是不可互换的。然后，我们证明了一类三角形的Fuchsian群是{it not/}格。这是第一个这样的三角形。最后，我们结束时，显示如何可以指定代数，即由一个明确的多项式在两个变量，黎曼曲面和全纯的形式，是一个简单的连接合理的多边形。以前，这些表面是已知的几何描述。作为一个例子，我们展示了正多边形中的台球和著名的代数方程x^n + y^n = 1$的费马曲线之间的联系。
We define, following Veech, the Fuchsian group $Gamma(P)$ of a rational polygon $P$. If $P$ is simply-connected, then ‘rational’ is equivalent to the condition that all interior angles of $P$ be rational multiples of $pi$. Should it happen that $Gamma(P)$ has finite covolume in $mathop{
m PSL}
olimits (2, {Bbb R})$ (and is thus a {it lattice}), then a theorem of Veech states that every billiard path in $P$ is either finite or uniformly distributed in $P$. We consider the Fuchsian groups of various rational triangles. First, we calculate explicitly the Fuchsian groups of a new sequence of triangles, and discover they are lattices. Interestingly, the lattices found are not commensurable with those previously known. We then demonstrate a class of triangles whose Fuchsian groups are {it not/} lattices. These are the first examples of such triangles. Finally, we end by showing how one may specify algebraically, i.e. by an explicit polynomial in two variables, the Riemann surfaces and holomorphic one-forms that are associated to a simply-connected rational polygon. Previously, these surfaces were known by their geometric description. As an example, we show a connection between the billiard in a regular polygon and the well-known Fermat curves of the algebraic equation $x^n + y^n = 1$.