Let's denote hyperbolic 3-space as $\text{H}_3 = \Bbb{C}\times (0,\infty)$. Then $\text{SL}(2,\mathbb{C})$ seems to act on $\text{H}_3$ via linear fractional transformations, i.e

$$\rho(M, z+wj) \mapsto M(z+wj) = \frac{\alpha(z+wj)+\beta}{\gamma (z+wj)+\delta} = z^{*}+w^{*} j \in \text{H}_{3}$$

Where

$$ z^{*}=\frac{(\alpha z+\beta)(\bar{\gamma} \bar{z}+\bar{\delta})+\alpha \bar{\gamma}w^{2}}{|\gamma z+\delta|^{2}+|\gamma|^{2} w^{2}} , \space\space w^{*}= \frac{w}{|\gamma z+\delta|^{2}+|\gamma|^{2} w^{2}}$$

Does this induce a group action on $\mathcal{L}^2(\text{H}_3)$ through the following map?

$$\rho(M, f)(z+wj)\mapsto f(M(z+wj))$$