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 $G_q\subset \mathrm{GL}(2,\mathbb{Z}[q^{\pm }])$ and then set ${\mathrm{PSL}}_q(2,\mathbb{Z}):=G_q/Z(G_q)$. We show that, for $\zeta \in {\mathbb{C}}^{\ast }$, ${\mathrm{PSL}}_q(2,\mathbb{Z}){|}_{q=\zeta }$ is finite, if and only if so is $G_q(\zeta ):=G_q{|}_{q=\zeta }\subset \mathrm{GL}(2,\mathbb{C})$, if and only if $\zeta ={\zeta }_n$ for $n=2,3,4,5$, where ${\zeta }_n$ is a primitive $n$-th root of unity. 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.

This is a substantially revised version of the previous one. The title has been changed, and the content has been substantially reorganized and expanded, with new results and applications added. 19 pages, comments welcome!