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Eta invariant and Selberg zeta function of odd type over convex co-compact hyperbolic manifolds
Guillarmou, Colin,Moroianu, Sergiu,Park, Jinsung Elsevier 2010 Advances in mathematics Vol.225 No.5
<P><B>Abstract</B></P><P>We show meromorphic extension and give a complete description of the divisors of a Selberg zeta function of odd type ZΓ,Σo(λ) associated to the spinor bundle <I>Σ</I> on an odd dimensional convex co-compact hyperbolic manifold Γ\<SUP>H2n+1</SUP>. As a byproduct we do a full analysis of the spectral and scattering theory of the Dirac operator on asymptotically hyperbolic manifolds. We show that there is a natural eta invariant η(D) associated to the Dirac operator <I>D</I> over a convex co-compact hyperbolic manifold Γ\<SUP>H2n+1</SUP> and that exp(πiη(D))=ZΓ,Σo(0), thus extending Millson's formula to this setting. Under some assumption on the exponent of convergence of Poincaré series for the group <I>Γ</I>, we also define an eta invariant for the odd signature operator, and we show that for Schottky 3-dimensional hyperbolic manifolds it gives the argument of a holomorphic function which appears in the Zograf factorization formula relating two natural Kähler potentials for Weil–Petersson metric on Schottky space.</P>
Adiabatic limit of the eta invariant over cofinite quotients of PSL(2, ℝ)
Loya, Paul,Moroianu, Sergiu,Park, Jinsung London Mathematical Society 2008 Compositio mathematica Vol.144 No.6
<B>Abstract</B><P>The eta invariant of the Dirac operator over a non-compact cofinite quotient of PSL(2,ℝ) is defined through a regularized trace following Melrose. It reduces to the standard definition in terms of eigenvalues in the case of a totally non-trivial spin structure. When the <I>S</I><SUP>1</SUP>-fibers are rescaled, the metric becomes of non-exact fibered-cusp type near the ends. We completely describe the continuous spectrum of the Dirac operator with respect to the rescaled metric and its dependence on the spin structure, and show that the adiabatic limit of the eta invariant is essentially the volume of the base hyperbolic Riemann surface with cusps, extending some of the results of Seade and Steer.</P>