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Just for fun, `\( \)` -> `\\\\( \\\\)` instead of `$ $`
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LSpice
LSpice
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As noted in the commentscomments, 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.

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.

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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Calvin McPhail-Snyder
Calvin McPhail-Snyder
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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.