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I'm looking for a generalization of the calculation of the hyperbolic volume and Chern–Simons invariant for $\operatorname{SL}(2,\mathbb C)$ representations in terms of the Rogers dilogarithm.

Recall that for a complex Lie group $G$, we have the Cheeger–Chern–Simons class $\hat c_2:H_3(G,\mathbb Z)\to\mathbb C/\mathbb Z$ (that's the group homology of $G$ in the discrete topology, with coefficients in $\mathbb Z$). If $G=\operatorname{SL}(2,\mathbb C)$, there is a nice explicit formula for this map, which is important in the calculation of hyperbolic volume. We'll now describe it briefly.

Recall that the pre-Bloch group $\mathcal P(\mathbb C)$ is the free abelian group on symbols $[z]$ for $z\in\mathbb C\setminus\{0,1\}$ modulo a certain "five-term relation". There is a map $\mathcal P(\mathbb C)\to\bigwedge^2\mathbb C$ given by $[z]\mapsto z\wedge(1-z)$, and we define the Bloch group $\mathcal B(\mathbb C)$ to be the kernel of this map. Then $\mathcal B(\mathbb C)$ is isomorphic to $H_3(\operatorname{SL}(2,\mathbb C),\mathbb Z)$ (ignoring small 2- or 3-torsion). There is a natural map from $\mathcal P(\mathbb C)$ to $\mathbb C$ given by the Rogers dilogarithm function (which satisfies the same five-term relation), and this restricted to $\mathcal B(\mathbb C)$ is exactly the Cheeger–Chern–Simons class. [I'm lying a bit here; actually we need to use the extended Bloch group due to Neumann.] This description of $H_3(\operatorname{SL}(2,\mathbb C),\mathbb Z)$ is related to the Suslin complex for $\operatorname{SL}(2,\mathbb C)$ acting on $\mathbb P^1(\mathbb C)$.

Question: are there similar explicit formulas for $\hat c_2$ for other groups $G$? At best, I'd like a formula which works for any semisimple $G$, including exceptional groups, but even just giving something that works for $G=\operatorname{SL}(n,\mathbb C)$ would be greatly appreciated!

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As noted in the comments, a paper [1] of Garoufalidis, D. Thurston, and Zickert answers this question for $\operatorname{SL}_n(\mathbb{C})$.

[1] Garoufalidis, Stavros; Thurston, Dylan P.; Zickert, Christian K., The complex volume of \\(\mathrm{SL}(n,\mathbb{C})\\)-representations of 3-manifolds, Duke Math. J. 164, No. 11, 2099-2160 (2015). ZBL1335.57034.

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    $\begingroup$ MathJax note: because escaping is fun, you can't use \( \) in MathJax, but instead \\\\( \\\\) (which is why we mostly use $ $ 😁). See my (wrongly titled) MMO post Having the MO Mathjax parser recognise \( \) is a regex away (whose title I mis-escaped just now, ha!). I edited accordingly. $\endgroup$
    – LSpice
    Feb 8 at 3:42

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