I've been studying number theoretic methods of classifying hyperbolic 3-manifolds for over a year now. In particular, there is are the trace field, invariant trace field, quaternion algebra, and invariant quaternion algebra of a hyperbolic 3-manifold, or equivalently, of the associated Kleinian group. There are various methods of understanding these things, for instance a quaternion algebra over a number field can be understood in terms of its ramification set.

Up until now, it has seemed that Colin Maclachlan, Alan Reid, and Walter Neumann pioneered these ideas starting in the late 80s. Tonight I was surprised to stumble across an article by Armond Borel called "Commensurability classes and volumes of hyperbolic 3-manifolds," from 1981, in which he is clearly aware of a number field and quaternion algebra associated to a commensurability class of Kleinian groups, as well as classification of the quaternion algebras via ramification. I was aware of some of his work about general arithmetic subgroups of algebraic groups, but had no idea that he had done anything focused specifically on this application.

So my question is, what is the origin of these ideas? I always feel that it is a good idea to read the original sources directly, whether I understand them or not.


The original extension of these invariants to general hyperbolic 3-manifolds was by Neumann and Reid.

I think they were motivated by the associated classification of arithmetic hyperbolic 3-manifolds. The general theory of arithmetic lattices was due to Borel and Harish-Chandra. I believe the specific case of orders in division algebras was first explored by Eichler, but I don't read German, so I can't verify. See the references in MacLachlan-Reid.

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The initial germs of what you're interested in was discovered by Bob Riley and Bill Thurston, going back to the late 1960's.

Here is Riley's account: http://arxiv.org/abs/1301.4601

And here is some commentary on his account: http://arxiv.org/abs/1301.4599

More systematic study of things like trace fields came later of course, but the understanding of these as invariants goes back to those two people. Mostow rigidity is of course the theorem that makes hyperbolic invariants actual topological invariants. The proof of Mostow rigidity relevant for knot and link exteriors is due to Marden and Prasad, in the early 1970's.

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    $\begingroup$ Except it is 1970s, not 60s. $\endgroup$ – Misha Apr 29 '14 at 15:32
  • $\begingroup$ I had read that Riley article before, and I was aware of Riley and Thurston's role in the origins of 3D hyperbolic topology as we currently know it. What I want to know specifically is: who was the first to define the trace field and quaternion algebra of a Kleinian group (or equivalently, of a hyperbolic 3-orbifold)? --and the same question for the invariant version of each of those. $\endgroup$ – j0equ1nn Apr 29 '14 at 18:57
  • $\begingroup$ Trace fields of representations would have been considered standard mathematics in the 1970s. So I imagine their first usage for 3-manifolds would have been more or less simultaneous with Mostow rigidity and greeted without fanfare. But perhaps someone else knows otherwise. $\endgroup$ – Ryan Budney Apr 29 '14 at 21:55
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    $\begingroup$ I have no idea accurate this is, but I thought that the current notion of an invariant trace field (as defined for instance in Maclachlan and Reid's book) dates back to a 1989 paper of Reid called "A note on trace fields of Kleinian groups" in which he gives an example showing that the trace field is not a commensurability class invariant and proves that what we now call the invariant trace field is a commensurability class invariant. $\endgroup$ – user1073 Apr 30 '14 at 4:15
  • $\begingroup$ @BenLinowitz That was my impression too. Since Reid doesn't credit anyone else with the term in the references, it probably was him who coined the term. I know he was also working closely with Walter Neumann and Colin Maclachlan at the time too. I get the feeling I have as much info as anyone else on this, but your answer below is informative so I'm going to "check" that one. The answer above of course makes sense but I think it's a little obvious. $\endgroup$ – j0equ1nn May 9 '14 at 17:56

I unfortunately have no idea when the ideas you refer to first appeared, but just wanted to point out a 1969 paper of Takeuchi ( "On some discrete subgroups of SL_2(R)" -- http://repository.dl.itc.u-tokyo.ac.jp/dspace/bitstream/2261/7331/2/jfs160107.pdf) in which he studies arithmetic Fuchsian groups. He refers to the basic construction of discrete groups from quaternion algebras over number fields (at least in his setting) as already being well-known. Since you can get Kleinian groups by modifying this construction ever so slightly, I would imagine that the ideas you are interested in were already known (in the sense of connecting certain discrete groups with orders in quaternion algebras) by the late 1960's.

I should also mention a 1980 paper of Vigneras: "Varietes Riemanniennes isospectrales et non-isometriques". In this paper she uses quaternion orders to construct isospectral but not isometric Riemannian n-manifolds for every n>=2 (which in dimensions 2 and 3 are hyperbolic). I believe that the way that people like Maclachlan and Reid study the length spectrum of an arithmetic manifold (via embeddings of quadratic orders into quaternion orders) originate from this paper.

One last thought - I'm not sure if it was part of your question, but the classification of quaternion algebras over number fields in terms of ramification goes way back. For instance, by the end of the 1930's it was already known how to classify central simple algebras of arbitrary dimension over a number field in terms of ramification sets. This is a consequence of the exact sequence of Brauer groups that appears in local class field theory. See for instance: http://en.wikipedia.org/wiki/Brauer_group#Brauer_group_and_class_field_theory

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  • $\begingroup$ That's a helpful observation. I know that Dickson defined generalized quaternion algebras some time around 1918, and that his interest was number theoretic. It's not surprising folks would have been using ramification to classify them by the 1930s. What I was hoping to learn was when these were recognized as topological invariants in hyp' 3-man's, and when their "invariant" counterparts were recognized as commensurability invariants. I think maybe the issue here is that I'm looking for a single source for something that really was a convergence of a lot of different things. $\endgroup$ – j0equ1nn May 9 '14 at 17:51

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