Cohomology of kleinian groups

Abstract

Let [unk] be a (nonelementary) Kleinian group and q >/= 2 an integer. The group [unk] acts in a natural way on the vector space II(2q-2) of complex polynomials in one variable of degree </= 2q - 2. One can thus form H(1)([unk],II(2q-2)), the first cohomology group of [unk] with coefficients in II(2q-2). There are essentially two ways of constructing cohomology classes. One construction originated with Eichler and has recently been extended by Ahlfors. Another construction is due to Bers. We show that for finitely generated [unk], every cohomology class pepsilon H(1)([unk],II(2q-2)) can be written uniquely (if one chooses an invariant union of components of [unk]) as a sum of a Bers cohomology class and an Eichler cohomology class. Similar decompositions are obtained for the subgroups of parabolic cohomology classes introduced by Ahlfors. Some information on the structure of H(1)([unk],II(2q-2)) for infinitely generated groups [unk] is also obtained.