Notes of Alan Reid’s Cambridge lecture 09-05-2017

Arithmetic of Dehn surgery points

Joint work with Ted Chinburg and Matthew Stover.

1. Invariants of hyperbolic 3 -manifolds

The key players are finite volume hyperbolic 3-manifolds. An example is the figure-eight complement, whose fundamental group is generated by two matrices

$\displaystyle \begin{array}{rcl} \begin{pmatrix} 1&1\\0&1\end{pmatrix},\quad\begin{pmatrix} 1&0\\\omega&1\end{pmatrix} \end{array}$

where ${\omega^3=1}$.

Remove a neighborhood of the end, the boundary is a 2-torus. Glue a solid torus instead (there are several ways, indexed by pairs of integers ${r}$). Thurston that the resulting closed manifold ${K(r)}$ is hyperbolic for all but finitely many ${r}$.

Associated to fundamental group ${\Gamma\subset PSl(2,{\mathbb C})}$ are

• A trace field ${k_\Gamma:={\mathbb Q}[trace(\gamma)\,;\,\gamma\in\Gamma]}$. Infinitesimal rigidity implies that ${k_\Gamma}$ is a number field.
• A quaternion algebra ${A_\Gamma:=k_\Gamma[\pi(\Gamma)]}$, where ${\pi:\Gamma\rightarrow Sl(2,{\mathbb C})}$ is the holonomy. It turns out to be 4-dimensional.

Note that ${A_\Gamma}$ is used to produce an example of a full tower of finite Galois covers of a hyperbolic manifold which are homology spheres.

2. Result

The snap software (a merge of snappy and pari) is convenient to design approximations and deformations of hyperbolic 3-manifolds. Experiments indicate that interesting ramification behaviour occurs among trace fields and quaternion algebras of manifolds obtained by Dehn surgery.

Theorem 1 Let ${K}$ be a hyperbolic knot. Suppose that its Alexander polynomial ${\Delta_K}$ satisfies

(*) Square roots of roots ${z}$ of ${\Delta_K}$ generate the same field over ${{\mathbb Q}}$ as ${z+\frac{1}{z}}$.

Let ${B_r}$ be the quaternion algebra of the Dehn surgery manifold ${K_r}$. Then there exists a finite set ${S}$ of primes such that every prime over ${k_\Gamma}$ over which some ${B_r}$ ramifies divides an element of ${S}$.

3. Arithmetic of the canonical curve

Thurston showed that in the character variety of a 1-cusped hyperbolic manifold group, there is canonical complex curve, which contains all Dehn surgeries. For the figure eight knot, it is an elliptic curve. For the ${(-2,3,7)}$-Pretzel knot, it is a rational curve.

Kronheimer and Mrowcka proved that every nontrivial knot has an irreducible ${SU(2)}$-representation. Reducible representations are also of interest. Thanks to a not so well known theorem by de Rham, they are related to roots of the Alexander polynomial.

Over the function field of the canonical curve, there is a quaternion algebra. Azumaya algebras generalize central simple algebras over fields, where fields are replaced with commutative local rings. They captures the local behaviour at a point of the canonical curve. It does not always exist. The criterion for its existence (Azumaya positivity) turns out to be (*).

The fact that Azumaya positivity implies the ramification property in the theorem is a theorem by Harari.

Conjecture. If ${K}$ is a hyperbolic ${L}$-space knot, it is never Azumaya positive.
${L}$-space is defined in terms of Heegard-Floer homology.