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

Published online by Cambridge University Press:  17 September 2025

Renaud Detcherry  and
Ramanujan Santharoubane
Renaud Detcherry
Affiliation:
Institut de Mathématiques de Bourgogne, UMR 5584, CNRS, Université Bourgogne Franche-Comté, F-21000 Dijon, France renaud.detcherry@u-bourgogne.fr
Ramanujan Santharoubane
Affiliation:
Laboratoire de Mathématique d’Orsay, UMR 8628, CNRS, Université Paris-Saclay, Bâtiment 307, 91405 Orsay Cedex, France ramanujan.santharoubane@universite-paris-saclay.fr
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Abstract

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 pth 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 pth 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 pth powers of Dehn twists.

Keywords

mapping class groupsquantum representationsJohnson filtrations

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Type
Research Article
Information
Compositio Mathematica , Volume 161 , Issue 7 , July 2025 , pp. 1778 - 1798
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© The Author(s), 2025. The publishing rights in this article are licensed to Foundation Compositio Mathematica under an exclusive licence

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