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

Abstract

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 $\mathbb{C}$.

arXiv comment46 pages, v4: added results on nonconvergence beyond NECM; added proofs for meromorphic continuation of derivatives of the Lerch transcendent; final version accepted for publication