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In this paper, we mainly study geometric constructions of thin Blaschke products B and reducing subspace problem of multiplication operators induced by such symbols B on the Bergman space. Considering such multiplication operators M B , we present a representation of those operators commuting with both M B and M * B . It is shown that for "most" thin Blaschke products B, M B is irreducible, i.e. M B has no nontrivial reducing subspace; and such a thin Blaschke product B is constructed. As an application of the methods, it is proved that for "most" finite Blaschke products φ, M φ has exactly two minimal reducing subspaces. Furthermore, under a mild condition, we get a geometric characterization for when M B defined by a thin Blaschke product B has a nontrivial reducing subspace.

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In this paper, we combine methods of complex analysis, operator theory and conformal geometry to construct a class of Type II factors in the theory of von Neumann algebras, which arise essentially from holomorphic coverings of bounded planar domains. One will see how types of such von Neumann algebras are related to algebraic topology of planar domains. As a result, the paper establishes a fascinating connections to one of the long-standing problems in free group factors. An interplay of analytical, geometrical, operator and group theoretical techniques is intrinsic to the paper.

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