The twisted Selberg trace formula and the Selberg zeta function for compact orbifolds

Abstract

We study elements of the spectral theory of compact hyperbolic orbifolds Γ\H n . We establish a version of the Selberg trace formula for non-unitary representations of Γ and prove that the associated Selberg zeta function admits a meromorphic continuation to C. Orbifolds, orbibundles and pseudodifferential operators2.1. Definitions and examples. We begin with an informal introduction to orbifolds and orbibundles.Definition 2.1. An orbifold O is a topological space such that for each p ∈ O there exists a neighbourhood U p , an open contractible set V p on R n , and a finite group Γ p acting on V p and satisfying U p = V p /Γ p . Definition 2.2. An orbifold is said to be global quotient if it is the orbit space of a manifold under a global action of a discrete (not necessarily finite) group.Definition 2.3. An orbibundle π : E → O is a map with the following properties. Let p ∈ O and U p , Γ p , V p be as above. Then there exists a representation of Γ p on R k such that the restriction of π to π −1 (U p ) is diffeomorphic to π ′ = (V p × R k )/Γ p → V p /Γ p , where the action of Γ p on V p × R k is the diagonal one.

Meromorphic continuation of Selberg zeta functions with twists having non-expanding cusp monodromy

“…Similar to Müller, she allows an additional unitary twist of the central elements in K. She shows convergence of the twisted (Ruelle and) Selberg zeta functions on certain half-planes in C. Taking advantage of Müller's twisted Selberg trace formula she then proves meromorphic continuability of the twisted Selberg zeta functions to all of C and provides a spectral interpretation of their singularities. Fedosova [14] generalizes these results to arbitrary (i. e., not necessarily torsion-free) cocompact lattices Γ in G.…”

“…Our second main result concerns the meromorphic continuability of the infinite product (1). As already mentioned above, prior to this paper, meromorphic continuability of Z Γ,χ has been known for unitary representations only (note that [45,14] consider odd-dimensional spaces only, thus they do not treat Fuchsian groups). For these investigations several methods (e. g. trace formulas, microlocal analysis, transfer operator techniques) have been employed and applied to various classes of (Γ, χ), resulting in alternative or complementary proofs of meromorphic extendability in different generality.…”

“…It is natural to ask to which extent these results can be generalized if in (1) the representation χ is allowed to be non-unitary. First investigations in this direction have been conducted by Müller [35], Monheim [33], Deitmar-Monheim [9,8], Deitmar [7], Spilioti [45] and Fedosova [14].…”

The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion

“…Therefore, the spectral theory of ∆ τ,ρ on X is the same as the spectral theory of its lift to the finite cover, acting on sections invariants under the action of the finite group of Deck transformations. Hence, by [15,Theorem 4.3], we have the following properties (see also for the manifold case the work of Müller [17], and for the orbifold case the work of Fedosova [6] and Shen [23,Section 7]):…”

Hypoelliptic Laplacian and twisted trace formula

“…Here, following the ideas in [17, Section 8], we will be concerned with the asymptotics of the equivariant Ray-Singer analytic torsion for Z. A similar problem has already been considered by Ksenia Fedosova [30] using methods of harmonic analysis on the hyperbolic spaces [31]. In this paper, we will exploit instead our explicit formula for twisted orbital integrals.…”