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:

Why did Müller's technique of relating the combinatorial and Hodge Laplacians not extend to the unimodular setting?

Is the orthogonal assumption essential to passing between the spectra of the combinatorial and Hodge Laplacians?

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

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.)