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.

4. Link to Heegard-Floer homology?

Conjecture. If {K} is a hyperbolic {L}-space knot, it is never Azumaya positive.

{L}-space is defined in terms of Heegard-Floer homology.

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About metric2011

metric2011 is a program of Centre Emile Borel, an activity of Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75005 Paris, France. See http://www.math.ens.fr/metric2011/
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