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Asymptotic Behavior of the Selberg Zeta Functions for Degenerating Families of Hyperbolic Manifolds

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Abstract

Let {M k } be a degenerating sequence of finite volume, hyperbolic manifolds of dimension d, with d = 2 or d = 3, with finite volume limit M ∞. Let \({Z_{M_{k}} (s)}\) be the associated sequence of Selberg zeta functions, and let \({{\mathcal{Z}}_{k} (s)}\) be the product of local factors in the Euler product expansion of \({Z_{M_{k}} (s)}\) corresponding to the pinching geodesics on M k . The main result in this article is to prove that \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) converges to \({Z_{M_{\infty}} (s)}\) for all \({s \in \mathbf{C}}\)with Re(s) > (d − 1)/2. The significant feature of our analysis is that the convergence of \({Z_{M_{k}} (s)/{\mathcal{Z}}_{k} (s)}\) to \({Z_{M_{\infty}} (s)}\) is obtained up to the critical line, including the right half of the critical strip, a region where the Euler product definition of the Selberg zeta function does not converge. In the case d = 2, our result reproves by different means the main theorem in Schulze (J Funct Anal 236:120–160, 2006).

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Authors and Affiliations

  1. Department of Mathematics, University of Sarajevo, Zmaja od Bosne 35, 71 000, Sarajevo, Bosnia and Herzegovina

    M. Avdispahić & L. Smajlović

  2. Department of Mathematics, The City College of New York, Convent Avenue at 138th Street, New York, NY, 10031, USA

    J. Jorgenson

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  1. M. Avdispahić
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  2. J. Jorgenson
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  3. L. Smajlović
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Correspondence to J. Jorgenson.

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Communicated by S. Zelditch

The first and third named authors are partly supported by grants from the Federal Ministry of Education and Research of Bosnia and Herzegovina.

The second named author acknowledges support from NSF and PSC-CUNY grants.

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Avdispahić, M., Jorgenson, J. & Smajlović, L. Asymptotic Behavior of the Selberg Zeta Functions for Degenerating Families of Hyperbolic Manifolds. Commun. Math. Phys. 310, 217–236 (2012). https://doi.org/10.1007/s00220-011-1408-5

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