Height estimates for Bianchi groups

Consider the geometry of the action of Bianchi groups $\mathrm{SL}(2, \mathcal{O}_d)$ on the hyperbolic space $\mathbb{H}^3$, where $\mathcal{O}_d$ is the ring of integers of the imaginary quadratic field $K=\mathbb{Q}(\sqrt{-d})$. We obtain, for some values of $d$, a height estimate $H(M) \leq c D(z,t)^9$, for some matrix $M$ that takes a given point $(z, t) \in \mathbb{H}^3$ into the fundamental domain of the Bianchi group. Here, $c$ is a constant that does not depend on the point and $D(z,t)$ is an explicit function of the coordinates of the initial point. This generalizes a lemma of Habegger and Pila about the action of the modular group on $\mathbb{H}^2$.