A fundamental domain of Ford type for some subgroups of the orthogonal group

Abstract

We initiate a study of the spectral theory of the locally symmetric space $X=\Gamma \backslash G/K$, where $G=SO(3,Complex)$, $\Gamma =SO(3,Z[i])$, $K=SO3$. 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 $Γ_Z=\SO(2,1)_Z$ 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 domains of $\Gamma$ and ${\Gamma }_Z$ so described. We then use the previous results compute the covolumes of of the lattices $\Gamma$ and ${\Gamma }_Z$ in $G$ and $G_R$.

119+ii Pages, 1 Figure, contains proofs of main results in "A fundamental domain of Ford type for $S{O}_3(Z[i])\backslash S{O}_3(C)/SO(3)$ and for $SO(2,1{)}_Z\backslash SO(2,1)/SO(2)$"