Height estimates for Bianchi groups

We study the action of Bianchi groups on the hyperbolic $ 3- $space $ \mathbb{H}^3 $. Given the standard fundamental domain for this action and any point in $ \mathbb{H}^3 $, we give an upper bound for the height of the unique element in the group which sends the point into the fundamental domain. The height is bounded by a polynomial function on some coordinates of the point whose degree does not depend on the Bianchi group. This generalizes a similar result of Habegger and Pila for the action of the Modular group on the hyperbolic plane. Our main theorem can be applied in the reduction theory of binary Hermitian forms with entries in the ring of integers of quadratic imaginary fields.