Determinants of twisted Laplacians and the twisted Selberg zeta function

TL;DR

This paper establishes a relation between the twisted Selberg zeta function and the twisted Laplacian on hyperbolic orbisurfaces, generalizing Sarnak's result and analyzing the asymptotic behavior of associated torsion constants.

Contribution

It introduces a novel proof relating the twisted Selberg zeta function to the twisted Laplacian and computes the torsion factor explicitly, extending previous results to non-trivial representations.

Findings

Derived a new relation between Z(s;ρ) and the twisted Laplacian.

Explicitly computed the torsion factor and its dependence on representation parameters.

Proved the asymptotic behavior of the torsion factor matches higher-dimensional Reidemeister torsion.

Abstract

Let $X$ be an orbisurface, meaning a compact hyperbolic Riemann surface possibly with a finite number of elliptic points, and let $X_1$ denote its unit tangent bundle. We consider the twisted Selberg zeta function $Z(s;\rho)$ associated to a representation $\rho: \pi_1(X_1) \to \text{GL}(V_\rho)$. We prove a relation between the twisted Selberg zeta function $Z(s;\rho)$ and the regularized determinant of the twisted Laplacian associated to $\rho$. These results can be viewed as a generalization of a result due to Sarnak who considered the trivial character. Yet our proof is different, as it is based on evaluation of the Laplace-Mellin type integral transformations. Going further, we explicitly compute the multiplicative constant, which we call the torsion factor, and express its dependence on parameters which determine the representation. We study the asymptotic behavior of the constant for a sequence of non-unitary representations introduced by Yamaguchi and prove that the asymptotic behavior of this constant as the dimension of the representation tends to infinity is the same as the behavior of the higher-dimensional Reidemeister torsion on $X_1$ (up to an absolute constant).