Non-variational global coordinates for Teichmüller spaces

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Abstract

Let Γ be a terminal, torsion free, (regular) b-group of type (p, n), 2p — 2 + n > 0. Maskit [M3] has observed that the deformation space T(Γ) is a model for the Teichmüller space T(p, n) of Riemann surfaces of finite analytic type (p, n) (because Γ represents a surface of type (p,n) on its invariant component and, in general, 2p — 2 + n thrice punctured spheres— the latter carry no moduli). He showed that the group Γ can be constructed from 3p — 3 + n terminal b-groups of type (1, 1) or (0, 4) (hence with a one dimensional deformation space). Each one dimensional Teichmüller space can be identified with U, the upper half plane—the Teichmüller space of the torus.