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Bounds on entries in Bianchi group generators

Upper and lower bounds are given for the maximum Euclidean curvature among faces in Bianchi's fundamental polyhedron for $PSL_2(O)$ in the upper-half space model of hyperbolic space, where $O$ is an imaginary quadratic ring of integers with discriminant $Δ$. We prove these bounds are asymptotically within $(\log |Δ|)^{8.54}$ of one another. This improves on the previous best upper-bound, which is roughly off by a factor between $Δ^2$ and $|Δ|^{5/2}$ depending on the smallest prime dividing $Δ$. The gap between our upper and lower bounds is determined by an analog of Jacobsthal's function, introduced here for imaginary quadratic fields.