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

Let K be a Q-Clifford algebra associated to an $(n-1)$-ary positive definite quadratic form and let O be a maximal order in K. A Clifford-Bianchi group is a group of the form SL₂(O) with O as above. The present paper is about the actions of SL₂(O) acting on hyperbolic space Hⁿ⁺¹ via M\"{o}bius transformations x↦ (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 Z-points of a Z-group scheme and are arithmetic subgroups of SO1,n+1(R)∘ with their M\"{o}bius action. We report on our findings concerning certain Clifford-Bianchi groups acting on H⁴, H⁵, and H⁶ .