Let $\mathcal{H}_2 = \{(x,t) \in \mathbf{R}^2: t > 0\}$ be the upper half-plane, and let $\mathcal{H}_3$ be the hyperbolic 3-space $\{(x,t) \in \mathbf{C} \times \mathbf{R}: t > 0\}$. Clearly $\mathcal{H}_2$ embeds in $\mathcal{H}_3$. There is an action of $PSL(2, \mathbf{C})$ on $\mathcal{H}_3$, extending the familiar action of $PSL(2, \mathbf{R})$ on $\mathcal{H}_2$.

If $\gamma \in PSL(2, \mathcal{O}_K)$, where $K$ is an imaginary quadratic field which is

not$\mathbf{Q}(\zeta_3)$, is it the case that we have must have either $\gamma \cdot \mathcal{H}_2 = \mathcal{H}_2$ or $\mathcal{H}_2 \cap \gamma \cdot\mathcal{H}_2 = \varnothing$?

For a general $PSL(2, \mathbf{C})$, or for non-integral $\gamma \in PGL(2, K)$, the intersection $\mathcal{H}_2 \cap \gamma \cdot\mathcal{H}_2$ can certainly be a nonempty proper subspace of $\mathcal{H}_2$ (in which case it's a geodesic arc). This case also seems to occur for some $\gamma \in PSL(2, \mathbf{Z}[\zeta_3])$, but I have found no other examples for any other quadratic fields. Is there a reason for this, or have I just not looked hard enough?