Analytic Torsion and R-torsion for Unimodular Representations

Let M be a closed COO -manifold of dimension n. Both R-torsion and analytic torsion are smooth invariants of acyclic orthogonal (or unitary) representations P of the fundamental group 1f l (M). The Reidemeister-Franz torsion (or R-torsion) r M(P) of P is defined in terms of the combinatorial structure of M given by its smooth triangulations. The analytic torsion T M(P) was introduced by Ray and Singer [RS] as an analytic counterpart of R-torsion. In order to define the analytic torsion one has to choose a Riemannian metric on M. Then T M(P) is a certain weighted alternating product of regularized determinants of the Laplacians on differential qforms of M with values in the flat bundle Ep defined by p. It was conjectured by Ray and Singer [RS] that T M(P) = r M(P) for all acyclic orthogonal (or unitary) representations p. This conjecture was proved independently by Cheeger [C] and the author [Mil]. The restriction to orthogonal (or unitary) representations is certainly a limitation of the applicability of this result if 1f1 (M) is infinite because an infinite discrete group will have, in general, many nonorthogonal finite-dimensional representations. It is the purpose of the present paper to remove this limitation. We call a representation P : 1fl (M) -+ GL(E) on a finite-dimensional real or complex vector space E unimodular if I detp(y)1 = 1 for all y E 1f1 (M). Then we define R-torsion and analytic torsion for unimodular representations, and the main result is that for odd-dimensional manifolds M the equality of the two torsions extends to all unimodular representations.