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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 $\mathcal P(\mathbb C)\to\wedge^2\mathbb C$ 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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Garoufalidis, D. Thurston, and Zickert are working on a related question. – Ian Agol Jul 21 '11 at 3:35
Any updates on this question? – user404153 Oct 22 '15 at 1:51

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