arXiv subjects

Polyxeni Spilioti

Publications and source records attributed to Polyxeni Spilioti.

12 recordsLinked to original sources

arXiv2026

Asymptotics of the spectral determinant of the weighted Laplacian for a sequence of compact Riemann surfaces of infinitely growing volume

Let $(X,\chi,k)$ be a triple consisting of a smooth, compact hyperbolic Riemann surface $X$ of genus $g$, and an $m$ dimensional unitary multiplier system $\chi$ of admissible weight $k$. Our first result establishes an analogue of the prime geodesic theorem for the weighted prime geodesic counting function associated to $(X,\chi,k)$. The error term we obtain is explicit with effectively computable constants which depend solely on the genus of $X$, the dimension of $\chi$, the length of shortest geodesic on $X$ and the smallest non-zero eigenvalues of the weighted Laplacian $\Delta_{2k}$ as well that of the scalar Laplacian $\Delta_{0}$. Our second result studies the asymptotic behavior of the spectral determinant $\det\Delta_{2k_n}$ for a sequence $(X_{n}, \chi_{n}, k_{n})$ for which the genus of $X_n$ tends to infinity. Under reasonably general circumstances, namely the existence of a weak spectral gap, a uniform discreteness of the underlying Fuchsian group, and a type of non-accumulation of bounded geodesics, we prove that $\log\det\Delta_{2k_n}/\mathrm{vol}(X_{n})$ converges to a constant $C_{\alpha}$ which depends only on $\alpha=\lim_{n\to\infty} k_n$. Our result is deterministic and is compatible with the three well-studied probabilistic models, namely Weil-Petersson, Brooks-Makover, and random covers model.

math.SP↗

arXiv2026

Twisted dynamical zeta functions and the Fried's conjecture

This is a survey article on the twisted dynamical zeta functions of Ruelle and Selberg and the Fried's conjecture. It is based on the mini-course: "Twisted Ruelle zeta function, complex-valued analytic torsion and the Fried's conjecture", given by the author during the thematic trimester programme: "Representation Theory and Noncommutative Geometry" at the Institut Henri Poincar\'{e}.

math.NT↗

arXiv2025

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 $Z(s;\rho)$ 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).

math.SP↗

arXiv2023

Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

For any pseudo-Riemannian hyperbolic space $X$ over $\mathbb{R},\mathbb{C},\mathbb{H}$ or $\mathbb{O}$, we show that the resolvent $R(z)=(\Box-z\operatorname{Id})^{-1}$ of the Laplace-Beltrami operator $-\Box$ on $X$ can be extended meromorphically across the spectrum of $\Box$ as a family of operators $C_c^\infty(X)\to \mathcal{D}'(X)$. Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in $\mathcal{D}'(X)$ forms a representation of the isometry group of $X$, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces. For Riemannian symmetric spaces analogous results were obtained by Miatello-Will and Hilgert-Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which $R(z)$ extends, and the residue representations can be infinite-dimensional.

math.SP↗

arXiv2022

Twisted Ruelle zeta function at zero for compact hyperbolic surfaces

Let $X$ be a compact, hyperbolic surface of genus $g\geq 2$. In this paper, we prove that the twisted Selberg and Ruelle zeta functions, associated with an arbitrary, finite-dimensional, complex representation $χ$ of $π_1(X)$ admit a meromorphic continuation to $\mathbb{C}$. Moreover, we study the behaviour of the twisted Ruelle zeta function at $s=0$ and prove that at this point, it has a zero of order $\dim(χ)(2g-2)$.

math.SP↗

arXiv2022

On the spectrum of twisted Laplacians and the Teichmüller representation

We consider Laplacians with non unitary twists acting on sections of flat vector bundles over compact hyperbolic surfaces. These non self-adjoint Laplacians have discrete spectrum inside a parabola in the complex plane. For representations of the fundamental group of the base surface which are of Teichmüller type, we investigate the high energy limit and give a precise description of the bulk of the spectrum where Weyl's law is satisfied in terms of critical exponents of the representations which are completely determined by the Manhattan curve associated to the Teichmüller deformation. Our main result provides a counting estimate for the eigenvalues outside the bulk with a polynomial improvement over Weyl's law.

math.SP↗

arXiv2021

The twisted Ruelle zeta function on compact hyperbolic orbisurfaces and Reidemeister-Turaev torsion

Let $X$ be a compact hyperbolic surface with finite order singularities, $X_1$ its unit tangent bundle. We consider the Ruelle zeta function $R(s;\rho)$ associated to a representation $\rho\colon\pi_1(X_1)\to\operatorname{GL}(V_\rho)$. If $\rho$ does not factor through $\pi_1(X)$, we show that the value at $0$ of the Ruelle zeta function equals the sign-refined Reidemeister-Turaev torsion of $(X_1, \rho)$ with respect to the Euler structure induced by the geodesic flow and to the natural homology orientation of $X_1$. It generalizes Fried's conjecture to non-unitary representations, and solves the phase and sign ambiguity in the unitary case. We also compute the vanishing order and the leading coefficient of the Ruelle zeta function at $s=0$ when $\rho$ factors through $\pi_1(X)$.

math.SP↗

arXiv2020

Twisted Ruelle zeta function on hyperbolic manifolds and complex-valued analytic torsion

In this paper, we study the twisted Ruelle zeta function associated with the geodesic flow of a compact, hyperbolic, odd-dimensional manifold $X$. The twisted Ruelle zeta function is associated with an acyclic representation $\chi\colon \pi_{1}(X) \rightarrow \GL_{n}(\C)$, which is close enough to an acyclic, unitary representation. In this case, the twisted Ruelle zeta function is regular at zero and equals the square of the refined analytic torsion, as it is introduced by Braverman and Kappeler in \cite{BK2}, multiplied by an exponential, which involves the eta invariant of the even part of the odd-signature operator, associated with $\chi$.

math.SP↗

arXiv2020

Functional equations of Selberg and Ruelle zeta functions for non-unitary twists

We consider the dynamical zeta functions of Selberg and Ruelle associated with the geodesic flow on a compact odd-dimensional hyperbolic manifold. These dynamical zeta functions are defined for a complex variable $s$ in some right-half plane of $\mathbb{C}$. In [Spi18], it was proved that they admit a meromorphic continuation to the whole complex plane. In this paper, we establish functional equations for them, relating their values at $s$ with those at $-s$. We prove also a determinant representation of the zeta functions, using the regularized determinants of certain twisted differential operators.

math.SP↗

arXiv2017

A prime Geodesic Theorem for SL3(Z)

We show a Prime Geodesic Theorem for the group SL3(Z), counting those geodesics whose lifts lie in the split Cartan subgroup. This is the first arithmetic Prime Geodesic Theorem of higher rank for a non-cocompact group.

math.NT↗

arXiv2015

Ruelle and Selberg zeta functions for non-unitary twists

In this paper, we study the Selberg and Ruelle zeta functions on compact hyperbolic odd dimensional manifolds. These zeta functions are defined on one complex variable $s$ in some right half-plane of $\mathbb{C}$. We use the Selberg trace formula for arbitrary not neccesarily unitary representations of the fundamental group to establish the meromorphic continuation of these zeta functions to the whole complex plane.

math.SP↗

arXiv2015

Twisted Dirac operators and dynamical zeta functions

In this paper, we consider the dynamical zeta functions of Ruelle and Selberg associated with the geodesic flow of a compact hyperbolic odd dimensional manifold $X$. These functions are initially defined on one complex variable $s$ in some right half-plane of $\mathbb{C}$. Our goal is the continue meromorphically the dynamical zeta functions to the whole complex plane, using the Selberg trace formula for arbitrary, not necessarily unitary, representations $χ$ of the fundamental group. First, we prove a trace formula for the integral operator $D^{\sharp}_χ(σ)e^{-t(D^{\sharp}_χ(σ))^{2}}$, induced by the twisted Dirac operator $D^{\sharp}_χ(σ)$ on $X$. Then we use these results to establish the meromorphic continuation of the dynamical zeta functions to $\mathbb{C}$.

math.SP↗