COHOMOLOGY OF KLEINIAN GROUPS

Let [unk] be a (nonelementary) Kleinian group andq ≥ 2 an integer. The group [unk] acts in a natural way on the vector space II2q —2of complex polynomials in one variable of degree ≤ 2q — 2. One can thus formH 1 ([unk],II2q —2), the first cohomology group of [unk] with coefficients in II2q —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 pεH 1 ([unk],II2q —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 ofH 1 ([unk],II2q —2) for infinitely generated groups [unk] is also obtained.