DISCRETE GROUPS GENERATED BY REFLECTIONS IN LOBAČEVSKIĬ SPACES
DISCRETE GROUPS GENERATED BY REFLECTIONS IN LOBAČEVSKIĬ SPACES
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LOBAČEVSKIĬ 空间中的反射生成的离散组
DOI：
10.1070/sm1967v001n03abeh001992
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发表时间：
1967
期刊：
影响因子：
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通讯作者：
E. Vinberg
中科院分区：
文献类型：
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作者：
E. Vinberg
众所周知，在欧几里德空间中，由反射生成的离散群由非负对称矩阵A=（θ i）指定，其中ai = 1，ai =-cos π/τ n ^（i F i），并且m i··可以假定值2，3，···，·Coxeter [11]给出了这些矩阵的完全枚举。本文的第一部分证明了（定理1）：在Lobacevskii空间中，由多次反射生成的具有有限基本区域的离散群可以借助于负惯性指数为1的对称矩阵来描述，这些对称矩阵具有aii= 1，oij =-cosn/rriij或at-<-1，并且包含足够多的（在某种意义上）Coxeter型子矩阵。这为我们提供了一个有效的手段来构建这样的群体。
It is well known that in Euclidean space discrete groups generated by reflections are specified by nonnegative symmetrical matrices A=(θ;;), where ai;= 1, a-=—cos π/τη^(i F/), and the m,·· can assume the values 2, 3,···,· A complete enumeration of these matrices was given by Coxeter [1 1. In the first part of the present paper we shall prove (Theorem 1) that in Lobacevskii space the discrete groups generated by finitely many reflections and having a finite fundamental region can be described with the aid of symmetric matrices of negative inertial index 1 that have aii= 1, oi;=-cos n/rriij or at-<-1 and contain sufficiently many (in a certain sense) submatrices of Coxeter type. This provides us with an effective means for constructing such groups.