Compact Quotient Manifolds of Domains in a Complex 3-Dimensional Projective Space and the Lebesgue Measure of Limit Sets

Abstract

We shall consider compact complex manifolds of dimension three which have subdomains in a three dimensional projective space as their unramified even coverings. Assume that the subdomains contain projective lines. Then any pair of such manifolds can be connected together complex analytically by an analogue of Klein combinations [K1]. In this paper, we shall prove a (weak) analogue of two results of B. Maskit [M] on the Lebesgue measures of the limit set of Kleinian groups (Theorems A and B). In section 1, we shall give definitions of terms and precise statement of our result. In section 2, we shall analyze the limit set by the same method as that of Maskit. Section 3 is devoted to proving Theorem 2.1. As a corollary to Theorem 2.1, we obtain Theorem A, which is an analogue of Combination Theorem I of [M]. In section 4, we shall prove Theorem B, which is an analogue of Combination Theorem II of [M].