Determinants of twisted Laplacians and the twisted Selberg zeta function

Let $X$ be a compact hyperbolic surface with finite order singularities and $X_1$ its unit

tangent bundle. We consider the twisted Selberg zeta function $Z(s; \rho)$ associated with

a representation $\rho: \pi_1(X_1) \to GL(V_\rho)$. In this talk, we will present recent results concerning a relation between the twisted Selberg zeta function $Z(s; \rho)$ and the regularized

determinant of the twisted Laplacian. The main tool we use is the Selberg trace formula.

If $X$ has no finite order singularities, we obtain as a corollary a corresponding relation.

These results can be viewed as an extension to the non-unitary twists case of the results

by Sarnak and Naud. This is joint work with Jay Jorgenson and Lejla Smajlovic.