Regularized Determinants of the Laplacian for cofinite kleinian groups with Finite-Dimensional Unitary Representations

Friedman, Joshua S.

doi:10.1007/s00220-007-0330-3

Cited by 6 publications

(9 citation statements)

References 32 publications

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“…For Y big enough the truncation M (Y ) of M at height Y is diffeomorphic to the so-called Borel-Serre compactification of M . Fixing t it is a consequence of the Selberg trace formula (see e.g [58]. for the case of functions and [87, Equation (5.5)] for the general case) thatM(Y ) tr e −t∆ k (x, x)dx ∼ k 0 log Y + c. Here, the notation A(Y ) ∼ B(Y ) means that A(Y ) − B(Y ) → 0 as Y → +∞,and k 0 and c are positive constants that depends on t. (In the case of 0-forms k 0 = h 2π +∞ 0 e −t(1+s 2 ) ds, where h is the number of cusps.)…”

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confidence: 99%

On the growth of $L^2$-invariants for sequences of lattices in Lie groups

et al. 2017

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Abstract. We study the asymptotic behaviour of Betti numbers, twisted torsion and other spectral invariants of sequences of locally symmetric spaces. Our main results are uniform versions of the DeGeorge-Wallach Theorem, of a theorem of Delorme and various other limit multiplicity theorems.A basic idea is to adapt the notion of Benjamini-Schramm convergence (BSconvergence), originally introduced for sequences of finite graphs of bounded degree, to sequences of Riemannian manifolds, and analyze the possible limits. We show that BS-convergence of locally symmetric spaces Γ\G/K implies convergence, in an appropriate sense, of the normalized relative Plancherel measures associated to L 2 (Γ\G). This then yields convergence of normalized multiplicities of unitary representations, Betti numbers and other spectral invariants. On the other hand, when the corresponding Lie group G is simple and of real rank at least two, we prove that there is only one possible BS-limit, i.e. when the volume tends to infinity, locally symmetric spaces always BSconverge to their universal cover G/K. This leads to various general uniform results.When restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of BS-convergence which in turn implies upper estimates on the rate of convergence of normalized Betti numbers in the spirit of Sarnak-Xue.An important role in our approach is played by the notion of Invariant Random Subgroups. For higher rank simple Lie groups G, we exploit rigidity theory, and in particular the Nevo-Stück-Zimmer theorem and Kazhdan's property (T), to obtain a complete understanding of the space of IRSs of G.

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confidence: 99%

On the growth of $L^2$-invariants for sequences of lattices in Lie groups

et al. 2017

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“…In [27], W. Müller generalized the idea of a regularized difference of heat traces to other settings. Following this approach, J. Friedman in [11] defined a regularized determinant of the Laplacian for any finite-volume three-dimensional hyperbolic orbifolds with finite-dimensional unitary representations, which he then related to special values of the Selberg zeta-function.…”

Section: Non-compact Hyperbolic Riemann Surfacesmentioning

confidence: 99%

“…Starting with the truncated heat kernel, one defines a regularized zeta function as the Mellin transform of the trace of the truncated heat kernel modulo the factor 1 Γ(s) . Variants of the second approach can be found in [4,5,11,25,26,27,30,31], and many others.…”

Section: Process Of Zeta Regularizationmentioning

confidence: 99%

The determinant of the Lax–Phillips scattering operator

Friedman¹,

Jorgenson²,

Smajlović³

2020

Annales De l'Institut Fourier

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Let M denote a finite volume, non-compact hyperbolic surface without elliptic points, and let B denote the Lax-Phillips scattering operator. Using the superzeta function approach due to Voros, we define a Hurwitz-type zeta function ζ ± B (s, z) constructed from the resonances associated to zI − [(1/2)I ± B]. We prove the meromorphic continuation in s of ζ ± B (s, z) and, using the special value at s = 0, define a determinant of the operators zI − [(1/2)I ± B]. We obtain expressions for Selberg's zeta function and the determinant of the scattering matrix in terms of the operator determinants. Résumé. -Soit M une surface hyperbolique non compacte à volume fini sans points elliptiques, et soit B l'opérateur de diffusion de Lax-Phillips. En utilisant l'approche due à Voros sur la fonction superzeta, nous définissons une fonction zêta de type Hurwitz ζ ± B (s, z) construite à partir des résonances associées à zI − [(1/2)I ± B]. Nous prouvons le prolongement méromorphe en le paramètre s de ζ ± B (s, z) et, en utilisant la valeur spéciale à s = 0, définissons un déterminant des opérateurs zI − [(1/2)I ± B]. Nous obtenons des expressions pour la fonction zêta de Selberg et le déterminant de la matrice de diffusion en termes de déterminants d'opérateurs.

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“…Of course, in the spirit of [36] (see also [2,13,15,16,18,31] for various generalizations in noncompact situations), Corollary 6.5 should be a direct consequence of a more general result of the form…”

Section: The Curvature Of the Determinant Line Bundlementioning

confidence: 99%

Some Index Formulae on the Moduli Space of Stable Parabolic Vector Bundles

Albin¹,

Rochon

2013

J. Aust. Math. Soc.

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Abstract. We study natural families of ∂-operators on the moduli space of stable parabolic vector bundles. Applying a families index theorem for hyperbolic cusp operators from our previous work, we find formulae for the Chern characters of the associated index bundles. The contributions from the cusps are explicitly expressed in terms of the Chern characters of natural vector bundles related to the parabolic structure. We show that our result implies formulae for the Chern classes of the associated determinant bundles consistent with a result of Takhtajan and Zograf.

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Regularized Determinants of the Laplacian for cofinite kleinian groups with Finite-Dimensional Unitary Representations

Cited by 6 publications

References 32 publications

On the growth of $L^2$-invariants for sequences of lattices in Lie groups

On the growth of $L^2$-invariants for sequences of lattices in Lie groups

The determinant of the Lax–Phillips scattering operator

Some Index Formulae on the Moduli Space of Stable Parabolic Vector Bundles

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