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Register a new userThe twisted Selberg trace formula and the Selberg zeta function for compact orbifolds
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Authors
Ksenia Fedosova
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We study elements of the spectral theory of compact hyperbolic orbifolds $Gamma backslash mathbb{H}^{n}$. We establish a version of the Selberg trace formula for non-unitary representations of $Gamma$ and prove that the associated Selberg zeta function admits a meromorphic continuation to $mathbb{C}$.
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Let G be a semisimple Lie group and H a uniform lattice in G. The Selberg trace formula is an equality arising from computing in two different ways the traces of convolution operators on the Hilbert space L^2(G/H) associated to test functions. In this paper we present a cohomological interpretation of the trace formula involving the K-theory of the maximal group C*-algebras of G and H. As an application, we exploit the role of group C*-algebras as recipients of higher indices of elliptic differential operators and we obtain the index theoretic version of the Selberg trace formula developed by Barbasch and Moscovici from ours.
We give an explicit formula for the second variation of the logarithm of the Selberg zeta function, $Z(s)$, on Teichmuller space. We then use this formula to determine the asymptotic behavior as $text{Re} (s) to infty$ of the second variation. As a consequence, for $m in mathbb{N}$, we obtain the complete expansion in $m$ of the curvature of the vector bundle $H^0(X_t, mathcal K_t)to tin mathcal T$ of holomorphic m-differentials over the Teichmuller space $mathcal T$, for $m$ large. Moreover, we show that this curvature agrees with the Quillen curvature up to a term of exponential decay, $O(m^2 e^{-l_0 m}),$ where $l_0$ is the length of the shortest closed hyperbolic geodesic.
This paper initiates a study into the contribution to the trace provided by the conjugacy classes.
Over the last few years Pohl (partly jointly with coauthors) developed dual `slow/fast transfer operator approaches to automorphic functions, resonances, and Selberg zeta functions for a certain class of hyperbolic surfaces $Gammabackslashmathbb{H}$ with cusps and all finite-dimensional unitary representations $chi$ of $Gamma$. The eigenfunctions with eigenvalue $1$ of the fast transfer operators determine the zeros of the Selberg zeta function for $(Gamma,chi)$. Further, if $Gamma$ is cofinite and $chi$ is the trivial one-dimensional representation then highly regular eigenfunctions with eigenvalue $1$ of the slow transfer operators characterize Maass cusp forms for $Gamma$. Conjecturally, this characterization extends to more general automorphic functions as well as to residues at resonances. In this article we study, without relying on Selberg theory, the relation between the eigenspaces of these two types of transfer operators for any Hecke triangle surface $Gammabackslashmathbb{H}$ of finite or infinite area and any finite-dimensional unitary representation $chi$ of the Hecke triangle group $Gamma$. In particular we provide explicit isomorphisms between relevant subspaces. This solves a conjecture by Moller and Pohl, characterizes some of the zeros of the Selberg zeta functions independently of the Selberg trace formula, and supports the previously mentioned conjectures.
We initiate the study of Selberg zeta functions $Z_{Gamma,chi}$ for geometrically finite Fuchsian groups $Gamma$ and finite-dimensional representations $chi$ with non-expanding cusp monodromy. We show that for all choices of $(Gamma,chi)$, the Selberg zeta function $Z_{Gamma,chi}$ converges on some half-plane in $mathbb{C}$. In addition, under the assumption that $Gamma$ admits a strict transfer operator approach, we show that $Z_{Gamma,chi}$ extends meromorphically to all of $mathbb{C}$.
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