How to get 3-manifold, Knots from Number Fields

Question

I'm reading a paper On the Torsion Jacquet-Langlands correspondence by Akshay Venkatesh and Frank Calegari.

Truthfully speaking I have no idea what Jacquet-Landlands is. I'm just trying to understand why there are knots in a paper on algebraic number theory and some of the players involved in that paper.

To keep matters simple, how do we pass between number fields and 3-manifolds and why is this beneficial? Poking around this paper, I find a version of the congruence groups: $$\Gamma_0(\mathfrak{n}) = \left\{ \left( \begin{array}{cc} a & b \\ c & d\end{array} \right) : \mathfrak{n}\big|\, c\right\} \subset PGL_2(\mathbf{O}_F)$$

where $\mathbf{O}_F$ is an order of a number field. We get a 3-manifold by quotienting hyperbolic 3-space: $\mathbb{H}^3/\Gamma_0(\mathfrak{n})$. Apparently, there's also another similar way to do it with quaternions.

They then proceed to look at look at some group-cohomology invariants of the group and then they use some spectral theory and the rest of paper mostly goes over my head. Well... we do get this:

However, along the way, we took many detours to explore related phenomena, some of which was inspired by the data computed for the first author by Nathan Dunfield. In view of the almost complete lack of rigorous understanding of torsion for (nonHermitian) locally symmetric spaces, we have included many of these results, even when what we can prove is rather modest.

The take-home message seems to be that we have constructed a large collection of infinite groups and actions on low-dimensional spaces of interest number theorists. Arithmetic lattices look like they play an important role.

I guess I'm trying to understand better how this connection between knots and number fields works and how Calegari and Venkatesh are using it to get a handle of the many invariants (which I may save for later questions).

Answer 1

I think that the connection between knots and number fields you may be imagining from reading that paper is rather specious. It's a theorem of Alan Reid that the only arithmetic hyperbolic knot complement is the figure eight knot complement. More generally, only finitely many Bianchi groups may admit covers which are link complements; this follows from non-vanishing results concerning the interior cohomology of such manifolds (for large enough discriminant, there are interior cohomology classes coming either from quadratic base change if you are a number theorist or from totally geodesic surfaces fixed by complex conjugation if you are a topologist).

The Whitehead link does occur in the paper you mention, but only in the following context: there are two specific arithmetic manifolds $W$ and $M$ discussed as examples which are obtained by Dehn surgery on the Whitehead link. By a theorem of Lickorish, all $3$-manifolds may be obtained by surgery on link complements, so this in itself has no particular relation to arithmetic. It is true that $W$ and $M$ have fairly simple desciptions as Dehn fillings of a fairly simple link, and this is somewhat related to arithmeticity --- in part because the resulting manifolds have particularly small volume. In particular, the manifold $W$ coming from $(5,2)$, $(5,1)$ surgery on the Whitehead link is the Weeks manifold, which, by a theorem of Gabai, Meyerhoff, and Milley, is the smallest volume orientable hyperbolic $3$-manifold. (In general, arithmetic manifolds seem to be "over"-represented in the small volume hyperbolic manifolds.)

To summarize, the main topic of interest in the paper you mention is the study of arithmetic hyperbolic $3$-manifolds, with a particular interest in their integral cohomology (and its surprising links to K-theory, Galois representations, and functoriality), but not really to the study of knots or links. If you want some background reading on these manifolds (written for topologists rather than number theorists), then Katie's suggestion (Machlachlan-Reid's book) is a good one.