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

Abstract

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 $\zeta \in {\mathbb C}^*$, $\operatorname{PSL}_q(2,{\mathbb Z})|_{q=\zeta}$ is finite, if and only if so is $G_q(\zeta):=G_q|_{q=\zeta} \subset \operatorname{GL}(2,{\mathbb C})$, 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, $G_q(\zeta_n) \cap \operatorname{SL}(2,\mathbb{C})$ 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).