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

Abstract

For an odd-dimensional oriented hyperbolic manifold with cusps and strongly acyclic coefficient systems we define the Reidemeister torsion of the Borel-Serre compactification of the manifold using bases of cohomology classes defined via Eisenstein series by the method of Harder. In the main result of this paper we relate this combinatorial torsion to the regularized analytic torsion. Together with results on the asymptotic behaviour of the regularized analytic torsion, established previously, this should have applications to study the growth of torsion in the cohomology of arithmetic groups. Our main result is established via a gluing formula and here our approach is heavily inspired by a recent paper of Lesch.

The torsion in symmetric powers on congruence subgroups of Bianchi groups

“…Next, as in [Pfa13], for each P j ∈ P Γ and for Y > 0 one can define the regularized analytic torsion T (F P j ;Γ (Y ), ∂F P j ;Γ (Y ); E ρ ) of F P j ;Γ (Y ) and the bundle E ρ | F P j ;Γ (Y ) , where one takes relative boundary conditions. For different Y , these torsions are compared by the following gluing formula.…”

“…Lemma 2.2. Let c(n) ∈ R be as in [Pfa13,equation 15.10]. Then for Y 1 > 0 and Y 2 > 0 one has We let X denote the Borel-Serre compactification of X and we let τ Eis (X; E ρ ) be the Reidemeister torsion of X with coefficients in E ρ , defined as in [Pfa13, section 9].…”

“…For simplicity, we assume that Γ is normal in Γ 0 . Then for the torsion T X 0 (X; E ρ ), the main result of [Pfa13] can be restated as follows.…”

“…We now recall the description of the canonical bases in the cohomology H * (X; Eρ m ) which are used to define the Reidemeister torsion τ Eis (X; Eρ m ). For more details, we refer to [Pfa13]. We shall use the notation of [Pfa13, section 8].…”

“…The asymptotic behaviour of the regularized analytic torsion in the finite volume case for a variation of the local system has already been studied by Müller and the first author in [MP12] using the Selberg trace formula. Thus we have to study the error term which occured in [Pfa13] in the comparison formula between analytic and Reidemeister torsion. It turns out that our study of the error term can be performed without any changes also in the higher dimensional situations.…”