On finite index subgroups of a universal group

TL;DR

This paper explores the classification of finite index subgroups of a universal orbifold group related to the Borromean rings, focusing on those generated by rotations, to understand their structure and properties.

Contribution

It initiates the classification of finite index subgroups of the universal group U, especially those generated by rotations, a problem previously considered difficult.

Findings

Identification of finite index subgroups generated by rotations

Partial classification of subgroups related to the universal group U

Insights into the structure of subgroups corresponding to 3-manifold quotients

Abstract

The orbifold group of the Borromean rings with singular angle 90 degrees, $U$, is a universal group, because every closed oriented 3--manifold $M^{3}$ occurs as a quotient space $M^{3} = H^{3}/G$, where $G$ is a finite index subgroup of $U$. Therefore, an interesting, but quite difficult problem, is to classify the finite index subgroups of the universal group $U$. One of the purposes of this paper is to begin this classification. In particular we analyze the classification of the finite index subgroups of $U$ that are generated by rotations.