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

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 \(\mathbb {C}\).