On the asynchronous rational group

TL;DR

This paper proves that the asynchronous rational group is simple and not finitely generated, extending these results to related subgroups like rational bilipschitz homeomorphisms.

Contribution

It establishes the simplicity and infinite generation of the asynchronous rational group and related subgroups, providing new insights into their algebraic structure.

Findings

The asynchronous rational group is simple.

The asynchronous rational group is not finitely generated.

Results apply to subgroups like rational bilipschitz homeomorphisms.

Abstract

We prove that the asynchronous rational group defined by Grigorchuk, Nekrashevych, and Sushchanskii is simple and not finitely generated. Our proofs also apply to certain subgroups of the rational group, such as the group of all rational bilipschitz homeomorphisms.