# Relatively hyperbolic knot groups

I am looking for references on the geometry of knot groups. For instance, I am interested in the following question:

When is a knot group relatively hyperbolic?

Hyperbolic knots are known to have a group which is relatively hyperbolic, but is the reciprocal true? Is the answer easier when restricted to alternating knots? And what about acylindrically hyperbolic knot groups?

The following theorem, which can be deduced from a combination theorem of Dahmani, is stated as Theorem 7.2.2 in our book 3-manifold groups .

Theorem: Let $N$ be a compact, orientable, irreducible 3-manifold with empty or toroidal boundary. Let $M_1,\ldots,M_k$ be the maximal graph-manifold pieces of the JSJ decomposition of $N$, let $S_1,\ldots,S_l$ be the tori in the boundary of $N$ that adjoin a hyperbolic piece and let $T_1,\ldots,T_m$ be the tori in the JSJ decomposition of $N$ that separate two (not necessarily distinct) hyperbolic pieces of the JSJ decomposition. The fundamental group of $N$ is hyperbolic relative to the set of parabolic subgroups $$\{H_i\}=\{\pi_1M_p\}\cup\{\pi_1 S_q\}\cup\{\pi_1 T_r\}~.$$

In particular, such a 3-manifold $M$ is non-elementarily relatively hyperbolic if and only if it is not a graph manifold (including the possibility of being Seifert fibred or a Sol manifold).

I believe that, historically, knots whose complements are graph manifolds were called hose knots, so the statement is that a knot complement group is non-elementarily relatively hyperbolic if and only if it's not a hose knot.

You also ask about acylindrical hyperbolicity, a weaker notion which merely asks for some acylindrical action on a hyperbolic space.

It turns out that non-geometric graph manifold groups always act acylindrically on the Bass-Serre tree of the JSJ decomposition (see Lemma 2.4 here), and so are acylindrically hyperbolic. So the fundamental group of a knot complement will be acylindrically hyperbolic unless the knot is Seifert fibred.

Finally, the remaining Seifert fibred knot complements (except for the trivial knot) all admit $\mathbb{H}^2\times\mathbb{R}$ geometry. In particular, after quotienting out a central infinite-cyclic subgroup, they act properly on the hyperbolic plane. The central $\mathbb{Z}$ acting trivially is the reason the corresponding action of the knot group on the hyperbolic plane fails to be acylindrical -- so even in this case, the groups are "nearly" acylindrically hyperbolic.

• Thank you for your detailed answer. Do there exist methods to compute JSJ decompositions of knot complements? – Seirios Jul 27 '17 at 15:37
• An algorithm using normal surfaces was described by Jaco and Tollefson: Algorithms for the complete decomposition of a closed 3-manifold. Illinois J. Math. 39 (1995), no. 3, 358–406. As far as I can tell this hasn't been implemented (in Regina regina-normal.github.io, say). But there are plenty of knot-theorists who perform these calculations all the time, so a focused question on MO might yield some useful answers. – HJRW Jul 28 '17 at 9:18

EDIT: Perhaps your question is best answered with another question - "relatively hyperbolic with respect to what kinds of subgroups?" I wrote my answer assuming that the subgroups of interest are the JSJ and peripheral tori. HJRW's answer further allows the fundamental groups of graph-manifold pieces of the JSJ decomposition. Here is what I originally wrote.

Suppose that $K$ is a satellite knot where both the companion knot $C$ and the pattern link $P$ are hyperbolic. Then $K$ is not hyperbolic itself, but $\pi_1(S^3 - K)$ is relatively hyperbolic with respect to the images of the peripheral tori of $C$ and $P$.

If $\Gamma$ is a torus link group then $\Gamma$ is weakly relatively hyperbolic with respect to its centre $Z(\Gamma)$, but not relatively hyperbolic with respect to $Z(\Gamma)$. I believe that this is an obstruction, in general.

So, one answer to your question is: A knot group is relatively hyperbolic with respect to its JSJ and peripheral tori if and only if all of the pieces of its JSJ decomposition are hyperbolic.

It certainly can happen that hyperbolic knot groups have other "relatively hyperbolic structures" (eg using a quasi-fuchsian surface subgroup). So perhaps we can use similar surfaces for a similar purpose in satellite knots.