Freeness and $S$-arithmeticity of rational M\"{o}bius groups

TL;DR

This paper develops computational methods to determine non-freeness of certain M"obius groups over rationals, using algorithms for Zariski dense groups and group presentations, revealing structural properties and finite index relations.

Contribution

It introduces a new computational approach to certify non-freeness of rational M"obius groups and computes their minimal finite index subgroups within $ ext{SL}(2,R)$.

Findings

Established criteria for non-freeness based on finite index in $ ext{SL}(2,R)$

Developed algorithms for computing presentations of $ ext{SL}(2,R)$

Identified minimal finite index subgroups containing the M"obius groups

Abstract

We initiate a new, computational approach to a classical problem: certifying non-freeness of ($2$-generator, parabolic) M\"{o}bius subgroups of $\mathrm{SL}(2,\mathbb{Q})$. The main tools used are algorithms for Zariski dense groups and algorithms to compute a presentation of $\mathrm{SL}(2, R)$ for a localization $R= \mathbb{Z}[\frac{1}{b}]$ of $\mathbb{Z}$. We prove that a M\"{o}bius subgroup $G$ is not free by showing that it has finite index in the relevant $\mathrm{SL}(2, R)$. Further information about the structure of $G$ is obtained; for example, we compute the minimal subgroup of finite index in $\mathrm{SL}(2,R)$ that contains $G$.