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

TL;DR

This paper characterizes when the specializations of the $q$-deformed modular group at roots of unity are finite, revealing they are finite only at specific roots and linking them to well-known finite groups.

Contribution

It provides a complete classification of the finiteness of specialized $q$-deformed modular groups at roots of unity and connects these groups to classical finite groups like the binary tetrahedral and icosahedral groups.

Findings

Finite groups occur only at roots of unity with orders 2, 3, 4, 5.

Specialized groups at these roots are isomorphic to known finite groups.

Groups at roots of unity with order 6 are infinite but exhibit mild properties.

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).