Determinants of twisted Laplacians and the twisted Selberg zeta function

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 $Z(s;ρ)$ associated to a representation $ρ: π_1(X_1) \to \text{GL}(V_ρ)$. We prove a relation between the twisted Selberg zeta function $Z(s;ρ)$ and the regularized determinant of the twisted Laplacian associated to $ρ$. 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).

💡 Research Summary

The paper studies the relationship between the twisted Selberg zeta function and the regularized determinant of a twisted Laplacian on a compact hyperbolic orbisurface, extending earlier results that were limited to the trivial (unitary) character.

Geometric setting.
Let (X=\Gamma\backslash\mathbb H^{2}) be a compact hyperbolic orbisurface, possibly with finitely many elliptic points, and let (X_{1}=\Gamma\backslash G) ((G=\mathrm{PSL}(2,\mathbb R))) be its unit‑tangent bundle, a Seifert‑fibered 3‑manifold. The fundamental group fits into the exact sequence
(1\to\mathbb Z\to\pi_{1}(X_{1})\to\Gamma\to1).
A finite‑dimensional complex representation (\rho:\pi_{1}(X_{1})\to\mathrm{GL}(V_{\rho})) is considered without assuming unitarity. The central element (u) (a full rotation of the fibre) satisfies (\rho(u)=e^{-i\pi m},I) with (m\in(-1,1]). For each elliptic conjugacy class (\gamma_{j}) of order (\nu_{j}) the eigenvalues of (\rho(\gamma_{j})) are explicitly described, leading to the auxiliary quantities (\alpha_{j}(\ell)) and (\tilde\alpha_{j}(\ell)) that encode the elliptic contribution.

Twisted Selberg zeta function.
The twisted Selberg zeta function is defined by
\