# Reconciling Sullivan's theorem with the hyperbolic structure of the Figure–8 knot complement

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?

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.

• Thank you @ThiKu for the answer. I would like to understand the statement about " ignoring the boundary components of genus < 2" a bit better. Does it mean that the conformal structure on each of the genus-1 boundary components (or more precisely the parabolic elements) is completely fixed even if the 3-manifold itself may have different hyperbolic structures in general? – Chao-Ming Jian Nov 24 '19 at 23:32
• I don‘t think so. It is just that a hyperbolic structure determines a unique conformal (euclidean) structure on the boundary torus, but not every euclidean structure on the torus will extend to a hyperbolic structure on the knot complement. In the infinite-volume case, when GF(M) is more than just a point, I actually don’t know whether the euclidean structure may vary with the hyperbolic structure. I‘m aware of some examples where it actually does not vary, perhaps this is a general phenomenon. – ThiKu Nov 25 '19 at 13:53
• Sorry, I should have made it clearer earlier. I meant to ask when the 3-manifolds M has a boundary \partial M that is a disjoint union of some genus-1 tori and other higher-genus surfaces, is the Teich(\partial M) appearing in the "Teich(\partial M) = GF(M)" referring to the space of the conformal structure on the higher-genus (g>1) components of \partial M? If yes, it seems to imply that the conformal structure on each of the genus-1 components of \partial M is fixed even if the hyperbolic structure of M may still vary. – Chao-Ming Jian Nov 25 '19 at 16:28
• Regarding your comment "In the infinite-volume case, when GF(M) is more than just a point, I actually don’t know whether the euclidean structure may vary with the hyperbolic structure. I‘m aware of some examples where it actually does not vary, perhaps this is a general phenomenon.", in fact, this is the very question I really want to understand initially :) Do you mind briefly explaining an example (or examples) that you have in mind? – Chao-Ming Jian Nov 25 '19 at 16:33
• The example I have in mind is any hyperbolic 3-manifold $M=M_1\cup M_2$, where $M_1$ and $M_2$ are glued along a totally geodesic surface $S$ (necessarily for genus $\ge 2$) and both have another boundary component, $M_1$ a torus, $M_2$ a surface $S_2$ of genus $\ge 2$. The Teichmüller space of $\partial M_2$ is just a product $T(\partial M_2)=T(S)\times T(S_2)$. So you can fix the hyperbolic structure on $S$ and now you just vary the hyperbolic structure on $M_2$. ... – ThiKu Nov 25 '19 at 20:18