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The slice determined by moduli equation \(x= \overline{y}\) in the deformation space of once punctured tori. (English) Zbl 0930.30038

The author gives another application of an algorithm developed in his earlier work on quasi-Fuchsian spaces of punctured tori. He uses as coordinates the traces \((x,y,z)\) of the generators of the group together with the trace of the product; these are related by the well-known Markov equation. The relation \(x= \overline y\) for groups in the quasi-Fuchsian space means that the surfaces represented by the corresponding group admit a symmetry: each is a rhombus. This subspace of quasi-Fuchsian space is called a “slice”. By clever computation, the author shows that if the parameters satisfy certain inequalities, the group is actually in the slice. His techniques don’t give any geometric intuition about why groups satisfying the inequalities are in the slice and hence they don’t say anything about where the boundary of the slice is. One would hope these techniques would be extended to determine the boundary.

MSC:

30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization)
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