Applications of the finite specializations of the $q$-deformed modular group

Abstract

Recently, Morier-Genoud and Ovsienko introduced the $q$-deformed modular group. For the 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. Although this result is essentially available in the existing literature, we give a precise proof for further developments. For example, we show that the set of traces of elements in $G_q(\zeta)$ and the set of the special values at $\zeta$ of the normalized Jones polynomials of rational links are finite if and only if $\zeta =\zeta_n$ for $n=2,3,4,5$ and also for $n=6$. We also study the quantized Pythagorean triples.