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

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