| 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). |
