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In the late 1970's, Cheeger and Müller independently proved the equality of analytic torsion and Reidemeister torsion for orthogonal representations, which had been conjectured by Ray-Singer. Their techniques were quite different: Cheeger used a surgery argument, whereas Müller approximated the Hodge Laplacian on differential forms by combinatorial Laplacians, using results of Dodziuk and Patodi.

In 1993, Müller extended this result from orthogonal to unimodular representations, essentially using Cheeger's surgery techniques.

My question is:

Do Müller's original methods involving the combinatorial Laplacian extend to the unimodular case, or is the orthogonal assumption necessary to use those methods?

(I would appreciate even heuristic comments relating to Müller's original paper or the combinatorial Laplacian techniques, since I am not familiar with them.)

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  • $\begingroup$ Are you sure that Müller's technique does not extend? Also, there is a more general result from Bismut and Zhang in Astérisque 205, 1992. Maybe because of that, Müller did not even try to extend his methods. $\endgroup$
    – Sebastian Goette
    Commented Nov 18, 2015 at 6:01
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    $\begingroup$ @Sebastian, no, I am not sure that Müller's technique does not extend, which is why I'm asking this question. I'm familiar with Bismut-Zhang, which of course also generalizes Müller's unimodular result (with a different proof). In any case, I'd like to know whether Müller did not even try to extend his methods, or if there is an essential obstruction. $\endgroup$
    – Phillip Andreae
    Commented Nov 18, 2015 at 6:10

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