Finiteness of specializations of the $q$-deformed modular group at roots of unity

Recently, Morier-Genoud and Ovsienko introduced the $q$-deformed modular group. For construction, they first gave a group $G_q\subset \mathrm{GL}(2,\mathbb{Z}[q^{\pm }])$ and then set ${\mathrm{PSL}}_q(2,\mathbb{Z}):=G_q\mathrm{/}Z(G_q)$. We show that for $\zeta \in {\mathbb{C}}^{\ast }$, ${\mathrm{PSL}}_q(2,\mathbb{Z}){\mathrm{\mid }}_{q=\zeta }$ is finite, if and only if so is $G_q(\zeta ):=G_q{\mathrm{\mid }}_{q=\zeta }\subset \mathrm{GL}(2,\mathbb{C})$, if and only if $\zeta ={\zeta }_n$ for $n=2,3,4,5$, where ${\zeta }_n$ is a primitive $n$-th root of unity. Moreover, $G_q({\zeta }_n)\cap \mathrm{SL}(2,\mathbb{C})$ 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).