$\Gamma(3+\omega)$ is a Kleinian group

The principal congruence subgroup $\Gamma(3+\omega)$ of the Bianchi group $\mathrm{SL}_2(\mathbb{Z}[\omega])$, $\omega = e^{2\pi i/3}$, acting on hyperbolic $3$-space by Möbius transformations, is a Kleinian group in the sense of the mission bundle: it acts by isometries preserving the hyperbolic volume, freely, and properly discontinuously. Proper discontinuity holds for the whole Bianchi group, because elements moving a compact set to meet itself have entries in a bounded disc, which contains finitely many Eisenstein integers. Freeness uses the level: $3+\omega$ is a prime of norm $7$, at a fixed point the trace is a rational integer in $[-2,2]$ congruent to $2$ modulo $3+\omega$, hence equal to $2$, which forces the identity. The quotient $\mathbb{H}^3/\Gamma(3+\omega)$ is therefore a hyperbolic $3$-manifold.