$SO(3,Z[i])\backslash SO(3,C)/SO(3)$和$SO(2,1)_Z\backslash SO(2,1)/SO(2)$的Ford型基本区
A fundamental domain of Ford type for $SO(3,Z[i])\backslash SO(3,C)/SO(3)$, and for $SO(2,1)_Z\backslash SO(2,1)/SO(2)$
摘要：
令$G=SO(3,C)$，$Γ=SO(3,Z[i])$，$K=SO(3)$，并令$X$是局部对称空间$Γ\backslash G/K$。在本文中，我们写下明确的方程定义一个基本区的$Γ$对$G/K$的作用。该基本区非常适合于研究$G/K$上的$Γ$ -不变函数理论。给出了作用在对称空间$G_{R}/K_R$上的$Γ$的子群$Γ_Z=SO(2,1)_Z$的定义基本区的方程，其中$G_R$是$G$的分裂实型SO（2，1），$K_R$是它的极大紧子群SO（2）.我们制定一个简单的几何关系之间的基本区的$Γ$和$Γ_Z$所描述的。这些基本领域是面向$X$和嵌入子空间$X_R=Γ_Z\backslash G_R/K_R$的谱理论的详细研究。}
英文摘要：
Let $G=SO(3,C)$, $Γ=SO(3,Z[i])$, $K=SO(3)$, and let $X$ be the locally symmetric space $Γ\backslash G/K$. In this paper, we write down explicit equations defining a fundamental domain for the action of $Γ$ on $G/K$. The fundamental domain is well-adapted for studying the theory of $Γ$-invariant functions on $G/K$. We write down equations defining a fundamental domain for the subgroup $Γ_Z=SO(2,1)_Z$ of $Γ$ acting on the symmetric space $G_{R}/K_R$, where $G_R$ is the split real form SO(2,1) of $G$ and $K_R$ is its maximal compact subgroup SO(2). We formulate a simple geometric relation between the fundamental domain of $Γ$ and $Γ_Z$ so described. These fundamental domains are geared towards the detailed study of the spectral theory of $X$ and the embedded subspace $X_R=Γ_Z\backslash G_R/K_R$.}
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