Bounds on Entries in Bianchi Group Generators

Abstract

Abstract Upper and lower bounds are given for the maximum Euclidean curvature among faces in the Ford domain for $\text {PSL}_2({\mathcal{O}})$ in the upper-half space model of hyperbolic space, where ${\mathcal{O}}$ is an imaginary quadratic ring of integers with discriminant $\Delta $. We prove these bounds are asymptotically within $(\log |\Delta |)^{8.54}$ of one another. This improves on the previous best upper bound, which is roughly off by a factor between $\Delta ^2$ and $|\Delta |^{5/2}$ depending on the smallest prime dividing $\Delta $. The gap between our upper and lower bounds is determined by an analog of Jacobsthal’s function, introduced here for imaginary quadratic fields.