# Is there a criterion for a link complement to have a hyperbolic structure with finite volume

For many links in $S^3$, the link complement can be equipped with a Riemannian structure which is complete, of constant sectional curvature -1, and has finite volume (i.e., a hyperbolic structure with finite volume). However, not every link complement can be endowed with such a structure. Does anybody know if there is a criterion for a link to have a corresponding hyperbolic structure? Since this problem is fundamental to the definition of hyperbolic invariants of knots and links, I guess people have already studied it thoroughly. However, I couldn't find such results. Could you please help me? Thank you very much!

There is a topological criterion due to Thurston. Using the JSJ machine (and work of many others) this criterion can also be phrased algebraically. I'll essay these below. Please note that the situation is much simpler for knots. To answer your question most directly, here is the desired reference to Wikipedia.

Now, here is Thurston's criterion. (EDIT: exposition improved after reading Bruno Martelli's answer.)

Suppose that $L$ is the link and $X$ is the link complement. Suppose $\pi = \pi_1(X)$. We assume the following properties (and each property assumes the proceeding ones). $\newcommand{\ZZ}{\mathbb{Z}}$

• $L$ is not a split link. Equivalently, $X$ is contains no essential two-sphere. Equivalently, $\pi$ is not a free product.
• $L$ is not the unknot. Equivalently, $X$ contains no essential disk. Equivalently, $\pi$ is not $\ZZ$.
• $L$ has no component that is an "undisturbed satellite knot". Equivalently, $X$ contains no essential torus.
• $L$ is not a torus knot. Equivalently, $X$ contains no essential annulus. These last two topological properties are equivalent to $\pi$ not containing a copy of $\ZZ^2$.

Then $X$ admits a hyperbolic structure.

• Thank you so much! Your answer is really helpful. However I have another question: even if X admits a hyperbolic structure, does it always have finite volume? (Or, to put it in another way, is the hyperbolic volume of a link always finite?) I have this question because we need the uniqueness of the hyperbolic structure in order to define hyperbolic invariants, but the Mostow rigidity theorem only works when the volume is finite. Do you know any related results? I might be asking very basic questions... So thank you for your help and patience! Jan 2 '14 at 0:35
• Yes, hyperbolic link complements always have finite hyperbolic volume. (There is a special case when we discuss the unknot, but I will ignore that for the present.) Here is a proof sketch. First, the finitely many cusp neighborhoods $C_i$ of $X$ each contain finite volume. (This is a standard exercise - see for example Peter Scott's article in the Bulletin of the AMS.) Next, notice $X - \cup C_i$ is compact. So this also has finite volume and we are done. Jan 2 '14 at 8:24