Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds

2009/01/26 by Colin Guillarmou, Sergiu Moroianu, Guillarmou, Colin +3
Mathematics · #11F72 #11M36 #37C30 #58J52 #Advanced Algebra and Geometry #Differential Geometry (math.DG) #FOS: Mathematics #Geometry and complex manifolds #Holomorphic and Operator Theory #Spectral Theory (math.SP) #math.DG #math.SP #msc:11F72 #msc:11M36 #msc:37C30 #msc:58J52

paper · pdf · doi:10.48550/arxiv.0901.4082

36 pages

arxiv created 2009/01/26 · openalex publication_date 2009/01/26 · arxiv updated 2009/12/01 · openalex created_date 2019/06/27 · openalex updated_date 2026/07/28

Abstract

We show meromorphic extension and analyze the divisors of a Selberg zeta function of odd type Z_Γ,Σ^{\rm o}(λ) associated to the spinor bundle Σ on odd dimensional convex co-compact hyperbolic manifolds X:=Γ\backslash\hh²ⁿ⁺¹. We define a natural eta invariant η(D) associated to the Dirac operator D on X and prove that η(D)=(1)/(πi)log Z_Γ,Σ^{\rm o}(0), thus extending Millson's formula to this setting. As a byproduct, we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We also define an eta invariant for the odd signature operator and, under some conditions, we describe it on the Schottky space of 3-dimensional Schottky hyperbolic manifolds and relate it to Zograf factorization formula.

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