Bounds on entries in Bianchi group generators
Abstract
Upper and lower bounds are given for the maximum Euclidean curvature among faces in Bianchi's fundamental polyhedron for $PSL_2(O)$ in the upperhalf space model of hyperbolic space, where $O$ 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 upperbound, 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, which is introduced here for imaginary quadratic fields.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.06973
 Bibcode:
 2021arXiv211006973M
 Keywords:

 Mathematics  Number Theory;
 11F06;
 11Y40;
 20H05;
 20H10;
 11R11;
 11Y16;
 57K32
 EPrint:
 22 pages, 6 figures