$\Gamma(2+i)$ is a Kleinian group

The principal congruence subgroup $\Gamma(2+i)$ of the Picard group $\mathrm{SL}_2(\mathbb{Z}[i])$, 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. Isometry and volume preservation hold for all of $\mathrm{SL}_2(\mathbb{C})$. Proper discontinuity is inherited from the Picard group: a compact set lies in a box $t_0 \le t \le T$, $|q|^2 \le R$, and if $g$ carries a point of the box into the box then $|c q + d|^2 = t/t' \le T/t_0$ bounds the Gaussian integers $c$, $d$, and by the same argument for $g^{-1}$ also $a$ and $b$. Freeness is where the level matters: at a fixed point the trace of $g$ is real and lies in $[-2, 2]$, for $g \in \Gamma(2+i)$ it is congruent to $2$ modulo $2+i$, and the only such Gaussian integer in $[-2, 2]$ is $2$, which forces $g = 1$. The quotient $\mathbb{H}^3/\Gamma(2+i)$ is therefore a hyperbolic $3$-manifold; this is the second milestone's witness, isolated as a reusable statement.