Intercusp geodesics and the invariant trace field of hyperbolic 3-manifolds (Q2789884)

This note studies geometric parameters of non-compact hyperbolic three-manifolds of finite volume. The parameters are obtained from geodesic segments between the normalized cusps of the manifold, that are perpendicular to the boundary of the cusps. In the universal covering the normalized cusps lift to a family of horoballs, and the geodesic segments lift to minimizing segments between the horoballs. The parameters are computed from those lifted segments, either as complex lengths (intercusp parameters) or using the complex affine structure of horospheres (translation parameters).NEWLINENEWLINEThe authors prove that these parameters are elements of the invariant trace field of the manifold. Let us recall that the invariant trace field is the number field spanned by traces of a subgroup of the fundamental group, the subgroup generated by squares, and that it is a commensurability invariant.NEWLINENEWLINEFurthermore, the authors give sufficient conditions for a collection of arcs and associated parameters to generate the invariant trace field. In particular, for a tunnel number \(k\) manifold it is enough to choose \(3k\) parameters (\(k\) intercusp parameters and \(2 k\) translation parameters). For many hyperbolic link complements, this approach allows to compute the field from a link diagram. Families of links are considered, for instace for two-bridge knots one parameter is sufficient. The authors also provide an infinite family of closures of braids that need only three parameters.NEWLINENEWLINEIt is very pleasant to read this note, it contains nice geometric ideas and the exposition is very clear.