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

TL;DR

This paper studies the meromorphic continuation of Selberg zeta functions associated with geometrically finite Fuchsian groups and certain representations, establishing convergence and extension properties under specific conditions.

Contribution

It introduces the study of Selberg zeta functions with non-expanding cusp monodromy and proves their meromorphic extension under a transfer operator approach.

Findings

Zeta functions converge on some half-plane in the complex plane.

Under certain conditions, zeta functions extend meromorphically to all of a7.

Results apply to a broad class of geometrically finite groups and representations.

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 $\mathbb{C}$.