Periods of iterated integrals of holomorphic forms on a compact Riemann surface

Holomorphic forms are integrated iteratedly along paths in a compact Riemann surface $M$ of genus $g$, thus inducing a homomorphism from the fundamental group $\Gamma = {\pi _1}(M,{P_0})$ to a proper multiplicative subgroup $G$ of the group of units in $\widehat{T({\Omega ^{1 \ast }})}$, where ${\Omega ^1}$ denotes the space of holomorphic forms on $T$ is the complex dual of ${\Omega ^1}$, $T$ means the associated tensor algebra and 11$\hat{ }$'' means completion with respect to the natural grading. The associated homomorphisms from $\Gamma /{\Gamma ^{(n + 1)}}$ to $G/{G^{(n + 1)}}$ reduces to the classical case ${H_1}(M) \to {\Omega ^{1 \ast }}$ when $n = 1$. We show that the images of $\Gamma /{\Gamma ^{(n + 1)}}$ are always cocompact in $G/{G^{(n + 1)}}$ and are discrete for all $n \geqslant 2$ if and only if the Jacobian variety $J(M)$ of $M$ is isogenous to ${E^g}$ for some elliptic curve $E$ with complex multiplication.