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In 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 (ArXiv), 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 $G\to G^{\mathrm{alg}}$ induces an isomorphism $$ H^{i}(G^{\mathrm{alg}},M)\to H^{i}(G,M) $$ for every (finite-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?

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    $\begingroup$ I wasn't aware of this concept. The only thing I would suggest is to see if the proofs in section 3 of Grunwald-Jaikin-Zapirain-Zalesskii carry over to the pro-algebraic case. $\endgroup$ – Ian Agol Jul 9 '19 at 5:39
  • $\begingroup$ For theoretical context, the condition of algebraic goodness is equivalent to saying that $(K(G,1)\otimes k)^{sch}\cong K(G^{alg},1)$, where $(-\otimes k)^{sch}$ is Toen’s schematization functor. $\endgroup$ – Patrick Elliott Jul 9 '19 at 11:45
  • $\begingroup$ Is it the same concept as in this answer? math.stackexchange.com/a/1163063 $\endgroup$ – Ian Agol Jul 9 '19 at 14:39
  • $\begingroup$ Your reference says that it's easy (at least pure braids). It cites another paper, which proves a theorem about extensions of groups, which is surprisingly weak. It imposes the hypothesis of type $F$, whereas I'd expect to only need $FP_\infty$ (eg, finite groups). Moreover, it requires the kernel to be free, which is a very strong hypothesis. I don't see that it uses that hypothesis, except to reduce the Hochschild spectral sequence to an LES, but I'm not sure. $\endgroup$ – Ben Wieland Jul 11 '19 at 21:21

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