As we know the figure-8 ($4_1$) complement can be obtained by quotienting $\mathbb{H}^3$ with an arithmetic Kleinian group, which has index 12 inside $PSL(2,\mathcal{O}_3)$. The resulting complete hyperbolic manifold has a cusp corresponding to the torus boundary.

On the other hand, the "conformal boundary" (boundary at infinity) of $\mathbb{H}^3$ is the sphere at infinity $S^2_{\infty}$. For a Kleinian group $\Gamma$, the hyperbolic manifold $\mathbb{H}^3/\Gamma$ inherits a conformal boundary $\Lambda/\Gamma$, where $\Lambda$ is the domain of discontinuity ($S^2_{\infty}$ with the limit set of $\Gamma$ excluded).

Then **how is the empty "conformal boundary" of $4_1$ complement related to its cusp?** (I know that the conformal structure of the torus boundary is called "cusp shape"); i.e.,

**should both of them correspond to the $z=0$ "end" in Poincare half-space model?**(since the torus boundary results from the truncated vertices on $S^2_{\infty}$ of 2 ideal tetrahedra under developing map upon gluing).

**Update:** the limit set of $\Gamma$ here is the entire $S^2_{\infty}$ (Corollary 11.8 in Bonahon), so domain of discontinuity as well as the conformal boundary is $\emptyset$.

**Confusion:** according to section 5 of Thurston, the *entire* boundary of a Kleinian manifold $(\mathbb{H}^3\cup\Lambda)/\Gamma$ is $\Lambda/\Gamma$, which is empty for $4_1$ complement. Then what is the role of the torus boundary (cusp)?

P.S.: As a physicist, I may not have stated the definition of "conformal boundary" rigorously, but in my mind it is the boundary in the sense of AdS/CFT correspondence (in Euclidean signature, $AdS_3$ is $\mathbb{H}^3$).