Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Takuma Byakuno, Xin Ren, Kohji Yanagawa

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 $ζ\in {\mathbb C}^*$, $\operatorname{PSL}_q(2,{\mathbb Z})|_{q=ζ}$ is finite, if and only if so is $G_q(ζ):=G_q|_{q=ζ} \subset \operatorname{GL}(2,{\mathbb C})$, if and only if $ζ=ζ_n$ for $n=2,3,4,5$, where $ζ_n$ is a primitive $n$-th root of unity. Moreover, $G_q(ζ_n) \cap \operatorname{SL}(2,\mathbb{C})$ 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).

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

Abstract

Recently, Morier-Genoud and Ovsienko introduced the -deformed modular group. For construction, they first gave a group and then set . We show that for , is finite, if and only if so is , if and only if for , where is a primitive -th root of unity. Moreover, is isomorphic to the binary tetrahedral group (resp. the binary icosahedral group), if (resp. ). When , 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).
Paper Structure (8 sections, 21 theorems, 71 equations)

This paper contains 8 sections, 21 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The proof of Theorem \ref{['n-all']}
  4. The proof of Theorem \ref{['n-all']} for n=3
  5. The proof of Theorem \ref{['n-all']} for n=4
  6. The proof of Theorem \ref{['n-all']} for n=5
  7. The proof of Theorem \ref{['finite']}
  8. The proof of Theorem \ref{['TR-omega']} and related results

Key Result

Theorem 1.1

For $\zeta \in \mathbb C^*$, $G_q(\zeta)$ is a finite group if and only if $\zeta =\zeta_n$ for $n=2,3,4,5$.

Theorems & Definitions (40)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Proposition 2.1: Leclere & Morier-Genoud LM
  • Definition 2.2: MO1MOV
  • Proposition 2.3: c.f. RY
  • Theorem 2.4: Leclere & Morier-Genoud LM
  • Theorem 2.5: Morier-Genoud & Ovsienko MO1, Bapat, Becker & Licata BBL
  • Theorem 2.6: Classification of finite subgroups of $\operatorname{SL}(2,\mathbb C)$
  • Remark 3.1
  • ...and 30 more