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 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 C.