Bounds on entries in Bianchi group generators

TL;DR

This paper establishes asymptotically tight bounds on the maximum Euclidean curvature of faces in Bianchi's fundamental polyhedron, improving previous bounds significantly and introducing a new Jacobsthal-like function for imaginary quadratic fields.

Contribution

It provides new asymptotic bounds on Euclidean curvature in Bianchi groups and introduces a Jacobsthal analog for imaginary quadratic fields.

Findings

Bounds are asymptotically within $(\log |\Delta| ight)^{8.54}$ of each other.

Improves previous upper bounds by a factor related to the discriminant.

Introduces a Jacobsthal analog for imaginary quadratic fields.

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 upper-half 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 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.