Analytic torsion versus Reidemeister torsion on hyperbolic 3-manifolds with cusps

Abstract

For a non-compact hyperbolic 3-manifold with cusps we prove an explicit formula that relates the regularized analytic torsion associated to the even symmetric powers of the standard representation of SL 2 (C) to the corresponding Reidemeister torsion. Our proof rests on an expression of the analytic torsion in terms of special values of Ruelle zeta functions as well as on recent work of Pere Menal-Ferrer and Joan Porti. 1 2 JONATHAN PFAFFLet us now turn to the actual setup of this paper. We let X be a hyperbolic 3-manifold which is not compact but of finite volume. If G := SL 2 (C), K := SU(2), then X := G/K can be identified with the hyperbolic 3-space and there exists a discrete, torsion free subgroup Γ of G such that X = Γ\ X. One can identify Γ with the fundamental group of X. Throughout this paper, we assume that Γ satisfies a certain condition, which is formulated in equation (2.2) below. Let ρ be an irreducible finite-dimensional complex representation of G. Restrict ρ to Γ and let E ρ be the associated flat vector-bundle over X. One can equip E ρ with a canonical metric, called admissible metric. The associated Laplace operator ∆ p (ρ) on E ρ -valued p-forms has a continuous spectrum and therefore, the heat operator exp(−t∆ p (ρ)) is not trace class. So the usual zeta function regularization can not be used to define the analytic torsion. However, picking up the concept of the b-trace of Melrose, employed by Park in a similar context, in [MP2] we introduced the regularized trace Tr reg e −t∆p(ρ) of the operators e −t∆p(ρ) and in this way we extended the definition of the analytic torsion to the non-compact manifold X. These definitions will be reviewed in section 4 below. Let T X (ρ) denote the analytic torsion on X associated to ρ.The aim of the present article is to find a suitable generalization of the aforementioned Cheeger-Müller theorems to the specific non-compact situation of the hyperbolic 3-manifold X with cusps. For m ∈ 1 2 N we let ρ(m) denote the 2m-th symmetric power of the standard representation of SL 2 (C). Let X be the Borel-Serre compactifcation of X. We recall that X is a compact smooth manifold with boundary and that X is diffeomorphic to the interior of X. Moreover, X and X are homotopy-equivalent. Thus, every representation ρ := ρ(m) of G also defines a flat vector bundle E ρ over X. Now by our assumption (2.2) on Γ and [MePo1, Proposition 2.8], the cohomology H * (X, ρ) never vanishes. Thus in order to define the Reidemeister torsion of E ρ , one needs to fix bases in the homology H * (X, ρ). However, by [MP2, Lemma 7.3] the bundle E ρ is L 2 -acyclic and thus the metrics on X and E ρ do not give such bases. This fact is a significant difference to the situation on a closed manifold described above and causes additional difficulties. To overcome this problem, we use the normalized Reidemeister torsion which was introduced by Menal-Ferrer and Porti [MePo2]. Recall that the boundary of X is a disjoint union of finitely many tori T i . For each i fix a non-trivial cycle θ i ∈ H 1 (T i...

Analytic torsion and R-torsion of Witt representations on manifolds with cusps

“…Combined with [ARS14, Corollary 12.2], this gives a description of Ruelle zeta functions in terms of intersection Rtorsion. Recently there has been also an impressive sequence of papers by Müller and Pfaff [MP12,Pfa14b,MP13a,Pfa15,Pfa14a,Pfa17], see also [Rai12,Rai13], in which the Selberg trace formula is used to great effect in analyzing analytic torsion. The methods in these papers are closely tied to the algebraic structure of locally symmetric spaces.…”

A Gluing Formula for the Analytic Torsion on Hyperbolic Manifolds With Cusps

“…We remark that in our earlier paper [Pf2], a specific comparison between certain regularized analytic torsions and certain Reidemeister torsions was obtained for the non-compact, finite-volume 3-dimensional hyperbolic case. The proof used a completely different method and was based on an explicit evaluation of special values of Ruelle zeta functions and on a result of Menal-Ferrer and Porti [MePo].…”

“…The proof used a completely different method and was based on an explicit evaluation of special values of Ruelle zeta functions and on a result of Menal-Ferrer and Porti [39]. The method of [58] does not generalize to higher dimensions nor even to general strongly acyclic coefficient systems in the three-dimensional case. Also, the result of [58] only holds for the quotients of two torsions associated to different representations of the fundamental group, and it does not use Eisenstein cohomology classes as a basis.…”

Dynamical zeta functions for geodesic flows and the higher-dimensional Reidemeister torsion for Fuchsian groups

“…Wüller and J. Pfaff developed the study of the asymptotic behavior of the analytic torsion for odd dimensional hyperbolic manifolds of finite volume in [MP12,MP13]. Pfaff also introduced in [Pfa14] the normalized analytic torsion corresponding to the higher-dimensional Reidemeister torsion invariant introduced by Menal-Ferrer and Porti in [MFP14] and compared them in detail. H. Goda and L. Bénard, J. Dubois, M. Heusener and J. Porti provided the asymptotic behavior of the higher-dimensional Reidemeister invariants with the descriptions in terms of the twisted Alexander invariants.…”