A fundamental domain of Ford type for $SO(3,Z[i])\backslash SO(3,C)/SO(3)$, and for $SO(2,1)_Z\backslash SO(2,1)/SO(2)$

Abstract

Let $G=SO(3,C)$, $\Gamma=SO(3,Z[i])$, $K=SO(3)$, and let $X$ be the locally symmetric space $\Gamma\backslash G/K$. In this paper, we write down explicit equations defining a fundamental domain for the action of $\Gamma$ on $G/K$. The fundamental domain is well-adapted for studying the theory of $\Gamma$-invariant functions on $G/K$. We write down equations defining a fundamental domain for the subgroup $\Gamma_Z=SO(2,1)_Z$ of $\Gamma$ 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 $\Gamma$ and $\Gamma_Z$ so described. These fundamental domains are geared towards the detailed study of the spectral theory of $X$ and the embedded subspace $X_R=\Gamma_Z\backslash G_R/K_R$.}