Finiteness of specializations of the $q$-deformed modular group at roots of unity
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866917325077217280 |
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| author | Byakuno, Takuma Ren, Xin Yanagawa, Kohji |
| author_facet | Byakuno, Takuma Ren, Xin Yanagawa, Kohji |
| contents | Recently, Morier-Genoud and Ovsienko introduced the $q$-deformed modular group.
For construction, they first gave a group $G_q \subset \operatorname{GL}(2, {\mathbb Z}[q^{\pm}])$ and then set $\operatorname{PSL}_q(2,{\mathbb Z}):=G_q/Z(G_q)$. We show that for $ζ\in {\mathbb C}^*$, $\operatorname{PSL}_q(2,{\mathbb Z})|_{q=ζ}$ is finite, if and only if so is $G_q(ζ):=G_q|_{q=ζ} \subset \operatorname{GL}(2,{\mathbb C})$, if and only if $ζ=ζ_n$ for $n=2,3,4,5$, where $ζ_n$ is a primitive $n$-th root of unity. Moreover, $G_q(ζ_n) \cap \operatorname{SL}(2,\mathbb{C})$ is isomorphic to the binary tetrahedral group (resp. the binary icosahedral group), if $n=3,4$ (resp. $n=5$). When $n=6$, the groups are infinite, but still "mild". We also give several applications (e.g., the special values of the normalized Jones polynomials of rational links). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_08439 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Finiteness of specializations of the $q$-deformed modular group at roots of unity Byakuno, Takuma Ren, Xin Yanagawa, Kohji Quantum Algebra Combinatorics Geometric Topology Number Theory 11F06, 05A30, 57K14, 11A55 Recently, Morier-Genoud and Ovsienko introduced the $q$-deformed modular group. For construction, they first gave a group $G_q \subset \operatorname{GL}(2, {\mathbb Z}[q^{\pm}])$ and then set $\operatorname{PSL}_q(2,{\mathbb Z}):=G_q/Z(G_q)$. We show that for $ζ\in {\mathbb C}^*$, $\operatorname{PSL}_q(2,{\mathbb Z})|_{q=ζ}$ is finite, if and only if so is $G_q(ζ):=G_q|_{q=ζ} \subset \operatorname{GL}(2,{\mathbb C})$, if and only if $ζ=ζ_n$ for $n=2,3,4,5$, where $ζ_n$ is a primitive $n$-th root of unity. Moreover, $G_q(ζ_n) \cap \operatorname{SL}(2,\mathbb{C})$ is isomorphic to the binary tetrahedral group (resp. the binary icosahedral group), if $n=3,4$ (resp. $n=5$). When $n=6$, the groups are infinite, but still "mild". We also give several applications (e.g., the special values of the normalized Jones polynomials of rational links). |
| title | Finiteness of specializations of the $q$-deformed modular group at roots of unity |
| topic | Quantum Algebra Combinatorics Geometric Topology Number Theory 11F06, 05A30, 57K14, 11A55 |
| url | https://arxiv.org/abs/2603.08439 |