Let $\bar{X}$ be a compact Riemann surface of genus $g>0$. Let $X$ be $\bar{X}$ minus a finite set of points $\{a_1,\cdots,a_n\}$ ($n\geq 1$). Let $X^{(r)}$ be the configuration space of $r$ ordered distinct points on $X$: $$X^{(r)}=X^r-\bigcup_{i\neq j}\{(x_1,\cdots,x_r): x_i=x_j\}$$

(1) Is there an explicit description of the Lie algebra associated to the lower central series of the fundamental group of $X^{(r)}$ (=pure braid group of $r$ strands on $X$) in terms of generators and relations?

(2) Are there cases (i.e. specific curves) for which an explicit description of $H^1_{dR}(X^{(r)})$ is known?

Any relevant references will be appreciated. For (1), a paper of Bezrukavnikov addresses $X=\bar{X}$ case. A paper of Nakamura, Takao and Ueno studies the non-compact situation but works with the weight central series (as opposed to the lower central series). In (2), by an explicit description I mean an explicit basis of $H^1_{dR}(X^{(r)})$.

Let $\bar{X}$ be a compact Riemann surface of genus $g>0$. Let $X$ be $\bar{X}$ minus a finite set of points $\{a_1,\ldots,a_n\}$ ($n\geq 1$). Let $X^{(r)}$ be the configuration space of $r$ ordered distinct points on $X$: $$X^{(r)}=X^r-\bigcup_{i\neq j}\{(x_1,\ldots,x_r): x_i=x_j\}$$

(1) Is there an explicit description of the Lie algebra associated to the lower central series of the fundamental group of $X^{(r)}$ (=pure braid group of $r$ strands on $X$) in terms of generators and relations?

(2) Are there cases (i.e. specific curves) for which an explicit description of $H^1_{dR}(X^{(r)})$ is known?

Any relevant references will be appreciated. For (1), a paper of Bezrukavnikov addresses $X=\bar{X}$ case. A paper of Nakamura, Takao and Ueno studies the non-compact situation but works with the weight central series (as opposed to the lower central series). In (2), by an explicit description I mean an explicit basis of $H^1_{dR}(X^{(r)})$.