  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 show that there exists an element in the group
which sends the given point into the fundamental domain such that its height is
bounded by a quadratic function on the coordinates of the point. This
generalizes and establishes a sharp version of 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. We also show
that the asymptotic behavior of the number of elements in a fixed Bianchi group
with height at most $T$ is biquadratic in $T$.
