# Mathematics Colloquia and Seminars

Given an arithmetic subgroup $G$ of the isometry group of hyperbolic n-space $H^n$, one can consider the orbifold $H^n/G$. Hyperbolic 2- and 3-orbifolds are reasonably well-understood; for example, hyperbolic 3-orbifolds correspond to orders of split quaternion algebras and there are algorithms that make use of this structure to compute geometric invariants of the orbifolds such as their volume, numbers of cusps, and fundamental groups. However, already hyperbolic 4-orbifolds belong to untamed wilds. We shall examine this frontier by introducing a class of arithmetic groups that have many of the same properties as the Bianchi groups and for which we can compute some geometric invariants of the orbifolds via algebraic invariants of rings with involution.