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.



Origin of number theoretic invariants associated to hyperbolic 3-manifolds
The original extension of these invariants to general hyperbolic 3-manifolds was by [Neumann and Reid](http://www.math.columbia.edu/~neumann/preprints/nrarith.pdf).


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](http://www.ams.org/journals/bull/1961-67-06/S0002-9904-1961-10699-X/S0002-9904-1961-10699-X.pdf). 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.


