arXiv subjects

Léo Bénard

Publications and source records attributed to Léo Bénard.

8 recordsLinked to original sources

arXiv2026

Multiplicity of non-acyclic ${\rm SL}_2$-representations and L-functions of the odd-twisted Whitehead links

We study the divisor of the Reidemeister torsion on the variety of irreducible ${\rm SL}_2\mathbb{C}$-characters of certain knots and links, and provide a geometric interpretation of them. We focus in particular on the family of odd-twisted Whitehead links $W_{2n-1}$ and prove that these divisors have multiplicity two. Furthermore, we apply these results to the study of the $L$-functions of the universal deformations of representations over fields with characteristic $p>2$ of these link groups.

math.GT↗

arXiv2025

Combinatorial zeta functions counting triangles

In this paper, we compute special values of certain combinatorial zeta functions counting geodesic paths in the (n-1)-skeleton of a triangulation of a n-dimensional manifold. We show that they carry a topological meaning. As such, we recover the first Betti number and L2-Betti number of compact manifolds, and the linking number of pairs of null-homologous knots in a 3-manifold. The tool to relate the two sides (counting geodesics/topological invariants) are random walks on higher dimensional skeleta of the triangulation.

math.GT↗

arXiv2023

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 $R(s;ρ)$ associated to a representation $ρ\colonπ_1(X_1)\to\operatorname{GL}(V_ρ)$. If $ρ$ does not factor through $π_1(X)$, we show that the value at $0$ of the Ruelle zeta function equals the sign-refined Reidemeister-Turaev torsion of $(X_1, ρ)$ 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 $ρ$ factors through $π_1(X)$.

math.SP↗

arXiv2020

Twisted $L^2$-torsion on the character variety

We define a twisted $L^2$-torsion on the character variety of 3-manifold $M$ and study some of its properties. In the case where $M$ is hyperbolic of finite volume, we prove that the $L^2$-torsion is a real analytic function on a neighborhood of any lift of the holonomy representation.

math.GT↗

arXiv2020

Reidemeister torsion form on character varieties

In this paper we define the adjoint Reidemeister torsion as a differential form on the character variety of a compact oriented 3-manifold with toral boundary, and prove it defines a regular volume form. Then we show that the torsion form can vanish only at singular points of the character variety. In fact, if the singular point corresponds to a reducible character, we show that the torsion does not vanish under a generic hypothesis on the Alexander polynomial, else we relate the vanishing order with the type of singularity. Finally we consider the ideal points added after compactification of the character variety. We bound the vanishing order of the torsion by the Euler characteristic of an essential surface associated to the ideal point by the Culler-Shalen theory. As a corollary we obtain an unexpected relation between the topology of those surfaces and the topology of the character variety.

math.GT↗

arXiv2019

Torsion function on character varieties

In this paper we define the Reidemeister torsion as a rational function on the geometric components of the character variety of a one-cusped hyperbolic manifold M. We study its poles and zeros, and we deduce sufficient conditions on the manifold M for this function being non-constant.

math.GT↗

arXiv2019

Asymptotics of twisted Alexander polynomials and hyperbolic volume

For a hyperbolic knot and a natural number n, we consider the Alexander polynomial twisted by the n-th symmetric power of a lift of the holonomy. We establish the asymptotic behavior of these twisted Alexander polynomials evaluated at unit complex numbers, yielding the volume of the knot exterior. More generally, we prove the asymptotic behavior for cusped hyperbolic manifolds of finite volume. The proof relies on results of Müller, and Menal-Ferrer and the last author. Using the uniformity of the convergence, we also deduce a similar asymptotic result for the Mahler measures of those polynomials.

math.GT↗

arXiv2019

A multivariable Casson-Lin type invariant

We introduce a multivariable Casson-Lin type invariant for links in $S^3$. This invariant is defined as a signed count of irreducible $\operatorname{SU}(2)$ representations of the link group with fixed meridional traces. For 2-component links with linking number one, the invariant is shown to be a sum of multivariable signatures. We also obtain some results concerning deformations of $\operatorname{SU}(2)$ representations of link groups.

math.GT↗