On periods of certain Eichler integrals for Kleinian groups (Q1073217)

Let G be a non-elementary Kleinian group, and let D be a G-invariant union of components of G. It is known from the work of Ahlfors and of Kra that the so-called period map is injective on the space of holomorphic Eichler integrals of weight 1-q on D if G is finitely generated. The aim of the present paper is to generalize this result to the wider class of Kleinian groups. For that end the author introduces new classes of Eichler integrals with growth restrictions in terms of Hardy classes. He compares his notion of Eichler integrals with the Eichler integrals considered by Kra. The main result is Theorem 4: If D/G is a union of parabolic Riemann surfaces, then the period map is injective on the author's space of Eichler integrals. In addition, the author proves a relation between the periods of his Eichler integrals and fine limits of these integrals at the points of the Martin boundary of D. The special case of a Fuchsian group is considered in the last section.