arXiv subjects

Renaud Detcherry

Publications and source records attributed to Renaud Detcherry.

At least 19 recordsLinked to original sources

arXiv2026

On the density and surjectivity of $\mathbf{SO(3)}$-Witten-Reshetikhin-Turaev quantum representations

In this paper, we establish several new fundamental properties of $\mathrm{SO}(3)$-quantum representations $\rho_{p,g,\underline{\lambda}}\colon\mathrm{PMod} (\Sigma_{g,n})\longrightarrow \mathrm{PSU}_{d_{p,g,\underline{\lambda}}}$ of mapping class groups of surfaces, at prime-order roots of unity. We show that for any surface $\Sigma_{g,n}$ of genus $g\geq 3$, any number $n\geq 0$ of punctures, and any coloration $\underline{\lambda}$ of the punctures, $\rho_{p,g,\underline{\lambda}}$ has dense image in the projective unitary group $\mathrm{PSU}_{d_{p,g,\underline{\lambda}}}$, extending a landmark result of Larsen and Wang. Moreover, we show that the representations $\rho_{p,g,\underline{\lambda}}$ are surjective modulo any unramified maximal ideal of $\mathbb{Z}[\zeta_p]$, establishing an effective version of strong approximation for these representations. We also give several applications of our main results to residual finite simpleness of $\mathrm{PMod}(\Sigma_{g,n})$ (answering a question of Masbaum and Reid); to subnormal cores of some subgroups of $\mathrm{PMod}(\Sigma_{g,n})$; to realizability of congruence classes of quantum invariants; to embedding obstructions between $3$-manifolds; and to homological stability for mapping class groups with coefficients in $\mathrm{SO}(3)$-quantum representations.

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arXiv2026

Embeddability between 3-manifolds is not stable

We prove that given two compact oriented $3$-manifolds $N$ and $M,$ with $M$ satisfying only a mild hypothesis, there is a hyperbolic $3$-manifold $N'$ arbitrarily ``closely related'' to $N,$ and such that $N'$ does not embed in $M.$ For instance, as a weak version of our main theorem, if $M$ is a rational homology sphere then for any $k\geq 1$ the $3$-manifold $N'$ can be chosen to be $Y_k$-equivalent to $N.$ Our techniques rely on the construction of $3$-manifolds with complicated Frohman--Kania-Bartoszy\'nska ideals, using the strong approximation for $\mathrm{SO}_3$-Witten-Reshetikhin-Turaev quantum representations of mapping class groups of surfaces.

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arXiv2026

The Andersen-Masbaum-Ueno conjecture for the derived subgroup of the Johnson kernel

A conjecture of Andersen, Masbaum and Ueno states that for any compact oriented surface $\Sigma_{g,n}$ and any pseudo-Anosov $f\in \mathrm{Mod}(\Sigma_{g,n}),$ the matrix $\rho_r(f)$ has infinite order for any large $r,$ where $\rho_r$ is the $\mathrm{SO}(3)$-WRT quantum representation of the mapping class group $\mathrm{Mod}(\Sigma_{g,n})$ at a primitive $r$-th root of unity. We prove this conjecture for prime $r$ and any $f\in [J_2(\Sigma_{g,n}),J_2(\Sigma_{g,n})],$ where $J_2(\Sigma_{g,n})$ is the Johnson kernel.

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arXiv2026

Small Seifert 3-manifolds with non-reduced $\mathrm{SL}_2(\mathbb{C})$-character scheme

We complete the work started in previous work of the author and Kalfagianni and Sikora, and give a complete description of the $\mathrm{SL}_2(\mathbb{C})$-character scheme $\mathcal{X}(M)$ of all small Seifert $3$-manifolds $M$. We find that $\mathcal{X}(M)$ is reduced if and only if $M$ admits no exceptional abelian character, and that exceptional abelian character have multiplicity $2$ in $\mathcal{X}(M).$

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arXiv2025

An effective proof of finiteness for Kauffman bracket skein modules

We prove a version of the finiteness conjecture for Kauffman bracket skein modules of $3$-manifolds with boundary, which was introduced by the second author in \cite{Det21}. In particular our methods, which are constructive, give an alternative proof of Witten's finiteness conjecture for the Kauffman bracket skein modules of closed $3$-manifolds, which was originally proved in \cite{GJS19}. Moreover, as a corollary we show that the peripheral ideal of any link is non-empty, answering a question of Frohman, Gelca and Lofaro \cite{FGL02}.

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arXiv2025

Seifert cobordisms and the Chen-Yang volume conjecture

We study the large $r$ asymptotic behavior of the Turaev-Viro invariants $TV_r(M; e^{\frac{2\pi i}{r}})$ of 3-manifolds with toroidal boundary, under the operation of gluing a Seifert-fibered 3-manifold along a component of $\partial M$. We show that the Turaev-Viro invariants volume conjecture is closed under this operation. As an application we prove the volume conjecture for all Seifert fibered 3-manifolds with boundary and for large classes of graph 3-manifolds.

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arXiv2024

On torsion in the Kauffman bracket skein module of $3$-manifolds

We study Kirby problems 1.92(E)-(G), which, roughly speaking, ask for which compact oriented $3$-manifold $M$ the Kauffman bracket skein module $\mathcal{S}(M)$ has torsion as a $\mathbb{Z}[A^{\pm 1}]$-module. We give new criteria for the presence of torsion in terms of how large the $SL_2(\mathbb{C})$-character variety of $M$ is. This gives many counterexamples to question 1.92(G)-(i) in Kirby's list. For manifolds with incompressible tori, we give new effective criteria for the presence of torsion, revisiting the work of Przytycki and Veve. We also show that $\mathcal{S}(\mathbb{R P}^3# L(p,1))$ has torsion when $p$ is even. Finally, we show that for $M$ an oriented Seifert manifold, closed or with boundary, $\mathcal{S}(M)$ has torsion if and only if $M$ admits a $2$-sided non-boundary parallel essential surface.

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arXiv2024

Skein modules and character varieties of Seifert manifolds

We show that the Kauffman bracket skein module of a closed Seifert fibered 3-manifold $M$ is finitely generated over $\mathbb Z[A^{\pm 1}]$ if and only if $M$ is irreducible and non-Haken. We analyze in detail the character varieties $X(M)$ of such manifolds and show that under mild conditions they are reduced. We compute the Kauffman bracket skein modules for these $3$-manifolds (over $\mathbb Q(A)$) and show that their dimensions coincide with $|X(M)|.$

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arXiv2024

On kernels of homological representations of mapping class groups

We study the kernels of representations of mapping class groups of surfaces on twisted homologies of configuration spaces. We relate them with the kernel of a natural twisted intersection pairing: if the latter kernel is trivial then the representation is faithful. As a main example, we study the representations $\rho_{n}$ of $\mathrm{Mod}(\Sigma_{g,1})$ based on a Heisenberg local system on the $n$ points configuration space of $\Sigma_{g,1}$, introduced by Blanchet--Palmer--Shaukat, and some of their specializations. In the one point configuration case, or when the Heisenberg group is quotiented by an element of its center, we find kernel elements in the twisted intersection form. On the other hand, for $n>2$ configuration points and the full Heisenberg local system, we identify subrepresentations of subgroups of $\mathrm{Mod}(\Sigma_{g,1})$ with Lawrence representations. In particular, we find one of these subgroups, which is isomorphic to a pure braid group on $g$ strands, on which the representations $\rho_n$ are faithful.

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arXiv2023

Non-semisimple quantum invariants and abelian classical shadows

Using the $U_q^Hsl_2$ non-semisimple invariants of 3-manifolds at odd roots of unity, we construct maps on the Kauffman bracket skein module at roots of unity of order twice an odd number, having any possible abelian non central character as classical shadow.

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arXiv2023

On the kernel of $\mathrm{SO}(3)$-Witten-Reshetikhin-Turaev quantum representations

In this paper, we study the kernels of the $\mathrm{SO}(3)$-Witten-Reshetikhin-Turaev quantum representations $\rho_p$ of mapping class groups of closed orientable surfaces $\Sigma_g$ of genus $g.$ We investigate the question whether the kernel of $\rho_p$ for $p$ prime is exactly the subgroup generated by $p$-th powers of Dehn twists. We show that if $g\geq 3$ and $p\geq 5$ then $\mathrm{Ker} \, \rho_p$ is contained in the subgroup generated by $p$-th powers of Dehn twists and separating twists, and if $g\geq 6$ and $p$ is a large enough prime then $\mathrm{Ker} \, \rho_p$ is contained in the subgroup generated by the commutator subgroup of the Johnson subgroup and by $p$-th powers of Dehn twists.

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arXiv2023

Kauffman bracket skein modules of small 3-manifolds

The proof of Witten's finiteness conjecture established that the Kauffman bracket skein modules of closed $3$-manifolds are finitely generated over $\mathbb Q(A)$. In this paper, we develop a novel method for computing these skein modules. We show that if the skein module $S(M,\mathbb Q[A^{\pm 1}])$ of $M$ is tame (e.g. finitely generated over $\mathbb Q[A^{\pm 1}]$), and the $SL(2,\mathbb C)$-character variety is reduced, then the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ is the number of closed points in this character variety. This, in particular, verifies a conjecture in the literature that relates the dimension $\dim_{\mathbb Q(A)}\, S(M, \mathbb Q(A))$ to the Abouzaid-Manolescu $SL(2,\mathbb C)$-Floer theoretic invariants, for large families of 3-manifolds. We also prove a criterion for reduceness of character varieties of closed $3$-manifolds and use it to compute the skein modules of Dehn fillings of $(2,2n+1)$-torus knots and of the figure-eight knot. The later family gives the first instance of computations of skein modules for closed hyperbolic 3-manifolds. We also prove that the skein modules of rational homology spheres have dimension at least $1$ over $\mathbb Q(A)$.

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arXiv2022

Compatible pants decompositions for $\mathrm{SL}_2(\mathbb{C})$-representations of surface groups

For any irreducible representation of a surface group into $\mathrm{SL}_2(\mathbb{C})$, we show that there exists a pants decomposition where the restriction to any pair of pants is irreducible and where no curve of the decomposition is sent to a trace $\pm 2$ element. We prove a similar property for $\mathrm{SO}_3$-representations. We also investigate the type of pants decomposition that can occur in this setting for a given representation. This result was announced in a previous paper of the first and third named authors, motivated by the study of the Azumaya locus of the skein algebra of surfaces at roots of unity.

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arXiv2022

An embedding of skein algebras of surfaces into quantum tori from Dehn-Thurston coordinates

We construct embeddings of Kauffman bracket skein algebras of surfaces (either closed or with boundary) into localized quantum tori using the action of the skein algebra on the skein module of the handlebody. We use those embeddings to study representations of Kauffman skein algebras at roots of unity and get a new proof of Bonahon-Wong's unicity conjecture. Our method allows one to explicitly reconstruct the unique representation with fixed classical shadow, as long as the classical shadow is irreducible with image not conjuguate to the quaternion group.

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arXiv2021

Pants complex, TQFT and hyperbolic geometry

We introduce a coarse perspective on relations of the $SU(2)$-Witten-Reshetikhin-Turaev TQFT, the Weil-Petersson geometry of the Teichm\"uller space, and volumes of hyperbolic 3-manifolds. Using data from the asymptotic expansions of the curve operators in the skein theoretic version of the $SU(2)$-TQFT, we define the quantum intersection number between pants decompositions of a closed surface. We show that the quantum intersection number admits two sided bounds in terms of the geometric intersection number and we use it to obtain a metric on the pants graph of surfaces. Using work of Brock we show that the pants graph equipped with this metric is quasi-isometric to the Teichm\"uller space with the Weil-Petersson metric and that the translation length of our metric provides two sided linear bounds on the volume of hyperbolic fibered manifolds. We briefly discuss how these relations are interpeted from the view point of $SU(2)$-character varieties of 3-manifolds. We also obtain a characterization of pseudo-Anosov mapping classes in terms of asymptotics of the quantum intersection number under iteration in the mapping class group and relate these asymptotics with stretch factors. We also discuss how these results fit with a conjecture of Andersen, Masbaum and Ueno about quantum representations of mapping class groups.

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arXiv2021

A quantum obstruction to purely cosmetic surgeries

We present new obstructions for a knot K in S^3 to admit purely cosmetic surgeries, which arise from the study of Witten-Reshetikhin-Turaev invariants at fixed level. In particular, we strengthen a recent result of Hanselman, showing that if K has purely cosmetic surgeries then the slopes of the surgeries are of the form 1/5k except if the Jones polynomial of K evaluated at a 5-th root of unity is 1. For any odd prime r, we also give an obstruction for K to have a 1/k surgery slope with k coprime to r that involves the values of the first (r-3)/2 colored Jones polynomials of K at an r-th root of unity. We verify the purely cosmetic surgery conjecture for all knots with at most 17 crossings.

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arXiv2020

Cosets of monodromies and quantum representations

We use geometric methods to show that given any $3$-manifold $M$, and $g$ a sufficiently large integer, the mapping class group $\mathrm{Mod}(Σ_{g,1})$ contains a coset of an abelian subgroup of rank $\lfloor \frac{g}{2}\rfloor,$ consisting of pseudo-Anosov monodromies of open-book decompositions in $M.$ We prove a similar result for rank two free cosets of $\mathrm{Mod}(Σ_{g,1}).$ These results have applications to a conjecture of Andersen, Masbaum and Ueno about quantum representations of surface mapping class groups. For surfaces with boundary, and large enough genus, we construct cosets of abelian and free subgroups of their mapping class groups consisting of elements that satisfy the conjecture. The mapping tori of these elements are fibered 3-manifolds that satisfy a weak form of the Turaev-Viro invariants volume conjecture.

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arXiv2020

Growth of quantum 6j-symbols and applications to the Volume Conjecture

We prove the Turaev-Viro invariants volume conjecture for a "universal" class of cusped hyperbolic 3-manifolds that produces all 3-manifolds with empty or toroidal boundary by Dehn filling. This leads to two-sided bounds on the volume of any hyperbolic 3-manifold with empty or toroidal boundary in terms of the growth rate of the Turaev-Viro invariants of the complement of an appropriate link contained in the manifold. We also provide evidence for a conjecture of Andersen, Masbaum and Ueno (AMU conjecture) about certain quantum representations of surface mapping class groups. A key step in our proofs is finding a sharp upper bound on the growth rate of the quantum $6j-$symbol evaluated at $q=e^{\frac{2πi}{r}}.$

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