Analytic torsion and Reidemeister torsion of hyperbolic manifolds with cusps

Müller, Werner; Rochon, Frédéric

doi:10.1007/s00039-020-00536-2

“…In a forthcoming paper, Theorem 1.2 will be used to generalize the main result of [21] by Lück and Schick, in which we will show the equality of Ray-Singer analytic 𝐿 2 -torsion and topological 𝐿 2 -torsion for a large class of flat, unimodular bundles over finite-volume, hyperbolic manifolds, which are studied by several other authors as well [1,4,27,28].…”

Section: Topmentioning

confidence: 87%

An L 2 -Cheeger Müller theorem on compact manifolds with boundary

Waßermann¹

2022

Annales Mathématiques Blaise Pascal

1

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“…The Ray-Singer conjecture was proved true independently by Cheeger [15] and Müller [54]. Since then analytic torsion has found several applications in diverse areas such as the study of the growth of torsion homology of arithmetic groups [6], locally symmetric spaces [39,40,49,55] and mathematical physics [8,32].…”

Section: Introductionmentioning

confidence: 99%

Equivariant analytic torsion for proper actions

Hochs¹,

Saratchandran²

2022

Preprint

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We construct an equivariant version of Ray-Singer analytic torsion for proper, isometric actions by locally compact groups on Riemannian manifolds, with compact quotients. We obtain results on convergence, metric independence, vanishing for even-dimensional manifolds, a product formula, and a decomposition of classical Ray-Singer analytic torsion as a sum over conjugacy classes of equivariant torsion on universal covers. We do explicit computations for the circle and the line acting on themselves, and for regular elliptic elements of SO0(3, 1) acting on 3-dimensional hyperbolic space. Our constructions and results generalise several earlier constructions of equivariant analytic torsion and their properties, most of which apply to finite or compact groups, or to fundamental groups of compact manifolds acting on their universal covers. These earlier versions of equivariant analytic torsion were associated to finite or compact conjugacy classes; we allow noncompact conjugacy classes, under suitable growth conditions.

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“…Coincidentally, Müller and Rochon recently studied the non-L 2 version of Theorem A for representations ρ of G that are not fixed by the standard Cartan involution on G corresponding to K < G [19]. In contrast to the L 2 -case, they showed that the two resulting torsion elements differ by a term related to the cusps of Γ\ H n = M which contribute non-trivially to the twisted cohomology H * (M, Γ\E ρ ) [19,Theorem 1.1].…”

Section: Introductionmentioning

confidence: 99%

“…Coincidentally, Müller and Rochon recently studied the non-L 2 version of Theorem A for representations ρ of G that are not fixed by the standard Cartan involution on G corresponding to K < G [19]. In contrast to the L 2 -case, they showed that the two resulting torsion elements differ by a term related to the cusps of Γ\ H n = M which contribute non-trivially to the twisted cohomology H * (M, Γ\E ρ ) [19,Theorem 1.1]. In fact, the situation they face (and, consequently, the method they employ) is fundamentally different from ours, the main difference being that the bundle E ρ ↓ H n central to our investigation is homogeneous (cf.…”

Section: Introductionmentioning

confidence: 99%