I am interested in the 3-manifolds with hyperbolic structures from the physics (gravity) perspective. I encounter this paper <https://arxiv.org/pdf/hep-th/9812206.pdf> whose Eq. (9) mentions a theorem by Sullivan, which states that if a 3-manifold $M$ allows at least one hyperbolic structure, there is a 1-1 correspondence between hyperbolic structures on $M$ and conformal structures on $\partial M$.


I want to apply this theorem to the figure-8 knot complement $M=S^3- K$, where $K$ denotes the figure-8 knot. The boundary of $M$ is $\partial M = T^2$. Hence, its conformal structure can be parametrized by a complex number $\tau$ (modulo $SL(2,Z)$). 
**My question is does Sullivan's theorem mean that for any choice of $\tau$ there is a corresponding hyperbolic structure on $M$ (regardless of whether the volume of the $M$ is finite or not)?**


Thurston's note constructs a hyperbolic structure for the figure-8 knot complement $M=S^3- K$ by gluing two tetrahedra together. The $\tau$ parameter for the boundary conformal structure seems to be uniquely fixed (to be some third root of unit?) in this construction. And the hyperbolic volume of $M$ is finite. If the answer to my question is yes, does it mean that, if $\tau$ deviates from the values given by Thurston's construction, $M$ can still have a hyperbolic structure but the volume has to be infinite? If the answer to my question is no, then what exactly is the statement of Sullivan's theorem?



Reconciling Sullivan's theorem with the hyperbolic structure of the Figure–8 knot complement
The statement is only true if you 


* restrict to geometrically finite hyperbolic metrics (possibly of infinite volume)
* and ignore parabolic elements, which basically means that you ignore the boundary components of genus < 2.


For a precise statement you may look at Section 3.1 of <http://www.math.harvard.edu/~ctm/papers/home/text/papers/iter/iter.pdf> (I don’t know what‘s the original source for Sullivan‘s theorem.)


So in the case of the figure eight knot complement, there is only one hyperbolic structure. In this case, this already follows from Mostow-Prasad rigidity.


