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

Abstract

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 $\mathbb{C}$. In addition, under the assumption that $Γ$ admits a strict transfer operator approach, we show that $Z_{Γ,χ}$ extends meromorphically to all of $\mathbb{C}$.