On the growth of torsion in the cohomology of arithmetic groups

Müller, Werner; Pfaff, Jonathan

doi:10.1007/s00208-014-1014-x

Cited by 20 publications

(15 citation statements)

References 15 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…One such application is to study the growth of torsion in the cohomology of an arithmetic group Γ. This was carried out in [MM11,MP14] when Γ is cocompact and without elliptic elements.…”

Section: Introductionmentioning

confidence: 99%

On the asymptotics of the analytic torsion for compact hyperbolic orbifolds

Fedosova

2015

Preprint

View full text Add to dashboard Cite

show abstract

“…One such application is to study the growth of torsion in the cohomology of an arithmetic group Γ. This was carried out in [MM11,MP14] when Γ is cocompact and without elliptic elements.…”

Section: Introductionmentioning

confidence: 99%

On the asymptotics of the analytic torsion for compact hyperbolic orbifolds

Fedosova

2015

Preprint

View full text Add to dashboard Cite

show abstract

“…For instance, on hyperbolic manifolds, it was used by Fried [Fri86] to express the R-torsion in terms of a Ruelle zeta function, a dynamical quantity defined in terms of closed geodesics. More recently, it has been used by various authors [BV13,CV12,Mül12,MM13,MP13b,MP13a,MP14b,BMZ17] to study the torsion of the homology of arithmetic groups.…”

Section: Introductionmentioning

confidence: 99%

Resolvent, heat kernel and torsion under degeneration to fibered cusps

Albin¹,

Rochon²,

Sher³

2014

Preprint

View full text Add to dashboard Cite

Manifolds with fibered cusps are a class of complete non-compact Riemannian manifolds including many examples of locally symmetric spaces of rank one. We study the spectrum of the Hodge Laplacian with coefficients in a flat bundle on a closed manifold undergoing degeneration to a manifold with fibered cusps. We obtain precise asymptotics for the resolvent, the heat kernel, and the determinant of the Laplacian. Using these asymptotics we obtain a topological description of the analytic torsion on a manifold with fibered cusps in terms of the R-torsion of the underlying manifold with boundary. Contents 1. Introduction 1 2. Fibered cusp surgery metrics 8 Resolvent under degeneration 15 3. Pseudodifferential operator calculi 15 4. Resolvent construction 21 5. Projection onto the eigenspace of small eigenvalues 40 Heat kernel under degeneration 43 6. Surgery heat space 43 7. Solving the heat equation 48 Torsion under degeneration 64 8. The R-torsion on manifolds with boundary 64 9. The intersection R-torsion of Dar and L 2 -cohomology 68 10. Analytic torsion conventions 72 11. Asymptotics of analytic torsion 77 12. A Cheeger-Müller theorem for fibered cusp manifolds 83 Appendix A. Model cases: Euclidean Laplacians and Dirac operators 85 Appendix B. Geometric microlocal preliminaries 89 Appendix C. Proof of composition formula 92 References 98

show abstract

“…These spaces have been studied by Vaillant [Vai01] and are closely related to the hyperbolic manifolds with cusps, studied e.g. by M üller-Pfaff in [MuPf14a], [MuPf14b]. On the other hand, a second example comes from scattering manifolds, studied by Guillarmou and Sher [GuSh13], with g ↾ U = dx 2 + x 2 g N and H * (N, E) = 0.…”

Section: 4mentioning

confidence: 99%

“…In the setting of non-compact hyperbolic spaces, the original definition of analytic torsion does not make sense due to the continuous spectrum of the Hodge Laplacian. Still, a renormalized version of analytic torsion exists and the intricate algebraic structure of the hyperbolic space, equipped with a flat Hermitian vector bundle that corresponds to a canonical non-unitary unimodular representation of the fundamental group, allows for a deep analysis of the relation between the renormalized analytic and Reidemeister torsions by Pfaff [Pfa14] and M üller-Pfaff [MuPf14a,MuPf14b]. Renormalized analytic torsion has also been discussed in the setting of non-compact asymptotically conical (scattering) manifolds by Guillarmou and Sher in [GuSh13], though its relation to the intersection Reidemeister torsion is still an open question.…”

Section: Introductionmentioning

confidence: 99%

Cheeger–Müller theorem on manifolds with cusps

Vertman

2018

Math. Z.

View full text Add to dashboard Cite

We prove equality between the renormalized Ray-Singer analytic torsion and the intersection R-torsion on a Witt-manifold with cusps, up to an error term determined explicitly by the Betti numbers of the cross section of the cusp and the intersection R-torsion of a model cone. In the first step of the proof we compute explicitly the renormalized Ray-Singer analytic torsion of a model cusp in general dimension and without the Witt-condition. In the second step we establish a gluing formula for renormalized Ray-Singer analytic torsion on a general class of non-compact manifolds in any dimension that includes Witt-manifolds with cusps, but also scattering manifolds with asymptotically conical ends. In the final step, a Cheeger-Müller theorem on cusps follows by a combination of the previous explicit computation and the gluing formula.

show abstract

On the growth of torsion in the cohomology of arithmetic groups

Cited by 20 publications

References 15 publications

On the asymptotics of the analytic torsion for compact hyperbolic orbifolds

On the asymptotics of the analytic torsion for compact hyperbolic orbifolds

Resolvent, heat kernel and torsion under degeneration to fibered cusps

Cheeger–Müller theorem on manifolds with cusps

Contact Info

Product

Resources

About