According to Thurston's geometrization conjecture (now probably proven by Perelman) we can decompose 3 manifolds into geometric pieces, each with one of 8 geometries. With the classification problem solved for 7 of the geometries, the interesting and difficult case of hyperbolic geometry remains. Since these manifolds are K(π,1)'s they are classified up to homotopy equivalence by their fundamental groups. Of course it is not known what this rather interesting family of infinite groups is! Mostow-Prasad rigidity says that homotopy equivalence implies isometry. Thus any geometric invariant, such as the volume of a hyperbolic 3-manifold, is at the same time an invariant of the topology and the fundamental group. There are many fascinating ways to exploit this "bridge".

One is to realize the fundamental group of an (orientable) finite volume hyperbolic 3-manifold as a Kleinian group – namely the group of covering transformations of the manifold – with entries in PSL(2,ℂ). The traces squared of these matrices turn out to be algebraic and generate a number field called the "invariant trace field" of the manifold (or Kleinian group). This is a commensurability invariant of the group, i.e. it is unchanged on passing to finite index subgroups. Further commensurability invariants are obtained by taking the quaternion algebra over the invariant trace field spanned by the matrices themselves.

My work on "snap" with Craig Hodgson and Walter Neumann has been to extend the computer program SnapPea with computations in high-precision and algebraic numbers so as to compute these invariants. Using the classification of quaternion algebras over a number field we obtain powerful commensurability invariants for all Kleinian groups. For the so-called arithmetic Kleinian groups this gives a complete classification up to commensurability.