Abstract : Let M_Q be the moduli stack of hyperbolic (g,n) curves. Its arithmetic fundamental group G_{g,n}^ar has the universal monodromy representation on a pro-unipotent group G_{g,n}^ar -> Out Pi -> Out p, where Pi denotes the profinite completion of the fundamental group of a (g,n)-curve, and p its Malcev completion over Q_l. We introduce a version of weighted Malcev completion of G_{g,n}^ar, denoted by cG_{g,n}^ar, through which the above representation factors. We show that this group is an extension of the arithmetic part (known as the motivic Galois group) by the geometric part (known as the relative Malcev completion of the mapping class group).