We initiate the study of Selberg zeta functions Z Γ,χ for geometrically finite Fuchsian groups Γ and finite-dimensional representations χ with non-expanding cusp monodromy. We show that for all choices of (Γ, χ), the Selberg zeta function Z Γ,χ converges on some half-plane in C. In addition, under the assumption that Γ admits a strict transfer operator approach, we show that Z Γ,χ extends meromorphically to all of C.

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.

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We present the Laplace operator associated to a hyperbolic surface Γ \ H and a unitary representation of the fundamental group Γ, extending the previous definition for hyperbolic surfaces of finite area to those of infinite area. We show that the resolvent of this operator admits a meromorphic continuation to all of C by constructing a parametrix for the Laplacian, following the approach by Guillopé and Zworski. We use the construction to provide an optimal upper bound for the counting function of the poles of the continued resolvent.

Let Γ be a geometrically finite Fuchsian group and suppose that χ : Γ → GL(V ) is a finite-dimensional representation with non-expanding cusp monodromy. We show that the parabolic Eisenstein series for Γ with twist χ converges on some half-plane. Further, we develop Fourier-type expansions for these Eisenstein series.