The Basic Theory of Clifford-Bianchi Groups for Hyperbolic n-Space

Abstract: Let $K$ be a $\mathbb{Q}$-Clifford algebra associated to an $(n-1)$-ary positive definite quadratic form and let $\mathcal{O}$ be a maximal order in $K$. A Clifford-Bianchi group is a group of the form $\operatorname{SL}2(\mathcal{O})$ with $\mathcal{O}$ as above. The present paper is about the actions of $\operatorname{SL}_2(\mathcal{O})$ acting on hyperbolic space $\mathcal{H}^{n+1}$ via M\"{o}bius transformations $x\mapsto (ax+b)(cx+d)^{-1}$. We develop the general theory of orders exhibiting explicit orders in low dimensions of interest. These include, for example, higher-dimensional analogs of the Hurwitz order. We develop the abstract and computational theory for determining their fundamental domains and generators and relations (higher-dimensional Bianchi-Humbert Theory). We make connections to the classical literature on symmetric spaces and arithmetic groups and provide a proof that these groups are $\mathbb{Z}$-points of a $\mathbb{Z}$-group scheme and are arithmetic subgroups of $\operatorname{SO}{1,n+1}(\mathbb{R})^{\circ}$ with their M\"{o}bius action. We report on our findings concerning certain Clifford-Bianchi groups acting on $\mathcal{H}^4$, $\mathcal{H}^5$, and $\mathcal{H}^6$.