The ring of quasimodular forms for a cocompact group

We describe the additive structure of the graded ring (M) over tilde* of quasimodular forms over any discrete and cocompact group Gamma subset of PSL(2. R). We show that this ring is never finitely generated. We calculate the exact number of new generators in each weight k. This number is constant for k sufficiently large and equals dim(C)(I/I boolean AND (I) over tilde (2)) where I and (I) over tilde are the ideals of modular forms and quasimodular forms, respectively, of positive weight. We show that (M) over tilde* is contained in some finitely generated ring (R) over tilde* of meromorphic quasimodular forms with dim (R) over tilde (k) = O(k(2)), i.e., the same order of growth as (M) over tilde*. (C) 2007 Elsevier Inc. All rights reserved.