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 $mathbb{C}$.