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The side-pairing elements of Maskit’s fundamental domain for the modular group in genus two. (English) Zbl 1002.57039

In this paper a thorough study of the hyperbolic geometry of surfaces of genus two is given, its main object are certain constellations of four geodesic loops on such surfaces ”standard chains”, defined by a chain type intersection scheme of the four curves). Cutting the surface open along a standard chain results in a disk, thus defines a marking of the surface and a point in Teichmüller space. If each choice of geodesic loop is shortest possible then the standard chain is called minimal and the surface together with this marking is said to lie in the Maskit domain of Teichmüller space. In fact, by previous work of B. Maskit [Invent. Math. 126, 341-390 (1996; Zbl 0873.32018)], such data (on surfaces of arbitrary genus) can be used to identify finite sided fundamental polyhedra for the action of the modular group on the Teichmüller space of marked hyperbolic surfaces of a given genus. ”The special nature of genus two has made it more accessible than in higher genus and we are able to produce a more detailed picture of the domain and its side-pairing transformations. If the domain can be shown to satisfy certain basic topological criteria, according to a classical theorem of Poincaré, then this would give a set of geometrical generators and relations of the modular group.”
Thus the goal is to sudy the intersections of the translates of the Maskit domain. Applying an element of the mapping class group to a standard chain on a surface one obtains another standard chain on that surface, hence a nonempty intersection of the Maskit domain with a translate corresponds to the case where both standard chains are minimal. On a surface of genus two, the curves of a standard chain intersect in one of six Weierstraß points of the surface. The first result of the paper states that distinct loops in a pair of minimal chains are either disjoint or intersect at Weierstraß points; as a consequence, there are only finitely many side-pairing elements. The main result of the paper then says that any minimal standard pair is equivalent to a minimal standard pair on one of two special surfaces (one is the well-known genus two surface with maximal isometry group). It is shown how the main results can be used to give a full list of side-pairing elements of the Maskit domain. The proofs of the paper are based on a careful (and rather long and technical) case-by-case analysis, using length inequality results for systems of non-dividing geodesics. A short history about finding a fundamental domain for the action of the modular group on Teichmüller space is given at the end of the introduction of the paper.

MSC:

57M50 General geometric structures on low-dimensional manifolds

Citations:

Zbl 0873.32018
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