Let $G$ be a group. It is well-known that $H_1(G,\mathbb{Z})=G/[G,G]$. Also (at least up to torsion) $[G,G]/[G,[G,G]]=\Lambda^2H^1(G,\mathbb{Z})/H_2(G,\mathbb{Z})$ as explained, for example, [in this answer][1]

Denote by $(G_r)$ the lower central series of $G:G_0=G,G_{r+1}=[G,G_r]$. Can further quotients $G_r/G_{r+1}$ be expressed in terms of group (co)homology?

Moreover, if $G=\pi_1(X)$ where $X$ is a compact algebraic variety, quotients $G_r/G_{r+1}$ possess a Hodge structure [introduced][2] by Hain. Can these Hodge structures be expressed in terms of the Hodge structure on the cohomology of $X$?


  [1]: https://mathoverflow.net/a/87169/39304
  [2]: https://services.math.duke.edu/~hain/papers/bowdoin.pdf