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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 answerin this answer

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 by Hain. Can these Hodge structures be expressed in terms of the Hodge structure on the cohomology of $X$?

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

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 by Hain. Can these Hodge structures be expressed in terms of the Hodge structure on the cohomology of $X$?

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

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 by Hain. Can these Hodge structures be expressed in terms of the Hodge structure on the cohomology of $X$?

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Lower central series quotients in terms of (co)homology

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

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 by Hain. Can these Hodge structures be expressed in terms of the Hodge structure on the cohomology of $X$?