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I’ve been studying the group of $\mathbb{Z}$-points of the simply connected Chevalley–Demazure group scheme of type $E_7$, denoted $G_{\text{sc}}(E_7,\mathbb{Z})$; see Vavilov and Plotkin - Chevalley groups over commutative rings: I. Elementary calculations for reference. In particular, I am interested in its first three integral homology groups. I have been able to compute the first two homology groups using various results in the literature. However, I have not found anything that helps compute the third homology group. Is there a way to compute this using results from the literature? Maybe there is a more general procedure to apply to this case as was applied in Theo Johnson-Freyd's answer to H_3 of SL(n,Z) and SL(n,F_p)?

If we can’t determine the third homology, then what can we say about it; for example, can we determine if it has 3-torsion, or it’s order?

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  • $\begingroup$ I'm no help for this problem. I just add a remark that I know some physicists who might like to know the answer. Specifically, 4D N=8 supergravity has, apparently, an $E_{7(7)}(\mathbb{Z})$-symmetry that arises from its description as a superstring compactification. Probably this symmetry enhances to $E_{7(7)}(\mathbb{R})$ at large distances, but microscopically it's $E_{7(7)}(\mathbb{Z})$. And cohomology of symmetry groups is the home of anomalies, extensions, etc. See the intro to arxiv.org/pdf/2210.04911 for details. $\endgroup$
    – Theo Johnson-Freyd
    Commented Jul 15 at 12:34
  • $\begingroup$ @TheoJohnson-Freyd It’s funny that you mention this, because that’s precisely why I’m interested. $\endgroup$
    – Noah B
    Commented Jul 15 at 13:47

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