Fuzzy Subgroups and the Teichmüller Space

There exists a generalization of the Teichmüller space of a covering group. In this paper we combine this generalized Teichmüller space T (G) and any fuzzy subgroup A : G−→ F where G is a subgroup of the group consisting of such orientation preserving and orientation reversing Möbius transformations which act in the upper half-plane of the extended complex plane. A partially ordered set F = (F ,≤) consists of stabilizers of G and all of their intersections. After preliminaries we present two new results concerning this special case of fuzzy subgroups. These conclusions are then applied to the known theory of the parametrization of the generalized Teichmüller space . As consequence, the equivalence classes of fuzzy subgroups (with an equivalence relation) become the elements of T (G) where G is generated by a finite set of hyperbolic Möbius transformations. Let the number of the generators be n. Then there is an embedding ψ : T (G) −→ R3n−3 and therefore a homeomorphism T (G)−→ψ(T (G)). Through parametrization of T (G) ψ(A(G)) has 3n−3 real coordinates which are also the coordinates of A(G) up to identification. Keywords— Fuzzy subgroups, Möbius groups, Möbius transformations, compact Riemann surfaces, Teichmüller spaces.