Equivalent trace sets for arithmetic Fuchsian groups

TL;DR

This paper demonstrates that certain subgroups of the modular and Bianchi groups share the same trace set as the original groups, revealing new structural properties of these arithmetic Fuchsian groups.

Contribution

It introduces the existence of infinite families of finite index subgroups with identical trace sets to the original groups, including explicit examples for the modular group.

Findings

The modular group has infinite families of subgroups with the same trace set.

Various congruence subgroups of the modular group share this property.

Bianchi groups also possess subgroups with identical trace sets.

Abstract

We show that the modular group has an infinite family of finite index subgroups, each of which has the same trace set as the modular group itself. Various congruence subgroups of the modular group, and the Bianchi groups, are also shown to have this property. In the case of the modular group, we construct examples of such finite index subgroups.