In this question Are mapping class groups of orientable surfaces good in the sense of Serre?this question it is asked whether the mapping class groups are 'good' in relation to pro-finite completion. Helpfully, one of the answers gives a link to a proof that the braid groups are good groups.
In Champs Affines https://arxiv.org/pdf/math/0012219.pdf(ArXiv) Toen, Toën introduces an analogous definition of an algebraically good group (Definition 3.4.1, p 84). In particular, a discrete group $G$ is algebraically good if and only if the pro-algebraic completion morphism $G\to G^{alg}$$G\to G^{\mathrm{alg}}$ induces an isomorphism $$ H^{i}(G^{alg},M)\to H^{i}(G,M) $$$$ H^{i}(G^{\mathrm{alg}},M)\to H^{i}(G,M) $$ for every (finite dimensional-dimensional) $G$-representation $M$.
My first question is simple: Are the braid groups algebraically good? What about the pure braid groups?
A more interesting question is the following: Under what constructions is algebraic goodness preserved?
For instance, when is an extension of algebraically good groups good?
