On pairs of once-punctured tori  pp. 183-208

On pairs of once-punctured tori

By Troels Jørgensen

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This work is a detailed study of the space of quasifuchsian once punctured torus groups in terms of their Ford (isometric) fundamental polyhedra. The key is a detailed analysis of how the pattern of isometric planes bounding the polyhedra change as one varies the group.


One possible approach to “Kleinian groups” is to ask: “How do they look?” It makes sense when the groups have been associated with natural fundamental polyhedrons.

In the case of Fuchsian groups, satisfactory answers were known to Fricke [FK26], but generally the situation becomes rather complicated.

It is natural to restrict the considerations to finitely generated groups and, hence-forward, we shall do so, since, in many respects, the class of groups which cannot be generated by a finite number of Möbius transformations seems to be too extensive for general studies – see for instance the examples of Abikoff [Abi71], [Abi73].

In preparation for an intuitive treatment of Kleinian groups, Ahlfors' finiteness paper [Ahl65a] contains much of the ground material. Also, it indicates one direction in which the above question might be specified, for instance, in order to attack the problems related to the characteristics of the limit sets, namely, whether the set of limit points situated on the boundary of the Dirichlet fundamental polyhedron is finite. Ahlfors proved that it has zero area [Ahl65b].

Contributing techniques from 3-dimensional topology, Marden [Mar74] described those groups which have finite sided polyhedrons and observed that they are stable in the sense of Bers [Ber70b].