  This paper is an exposition of the completion of a modular group with respect
to its inclusion into SL_2(Q) and the connection with the theory of modular
forms and variations of mixed Hodge structure over modular curves. Among the
goals of this paper are to give a context to Manin's iterated Shimura integrals
(iterated integrals of modular forms) and to study relations in the "maximal
Eisenstein quotient" of the completion of SL_2(Z). In particular, we use a
computation of Terasoma to prove that Pollack's quadratic relations in the
derivation Lie algebra of a rank two free Lie algebra lift to this Eisenstein
quotient, and are thus motivic. We also construct the normal functions
associated to Hecke eigen cusp forms.
