The infinite simple group V of Richard J. Thompson: presentations by permutations

TL;DR

This paper introduces a permutation-based perspective on Richard Thompson's group V, providing intuitive explanations for its simplicity and new minimal presentations with fewer generators and relations.

Contribution

It offers a natural permutation-based description of V, connecting it to finite alternating groups, and develops simplified presentations for the group.

Findings

Permutation-based description of V as products of transpositions

New small presentations with fewer generators and relations

Insight into the simplicity of V through permutation perspective

Abstract

We show that one can naturally describe elements of R. Thompson's finitely presented infinite simple group $V$, known by Thompson to have a presentation with four generators and fourteen relations, as products of permutations analogous to transpositions. This perspective provides an intuitive explanation towards the simplicity of $V$ and also perhaps indicates a reason as to why it was one of the first discovered infinite finitely presented simple groups: it is (in some basic sense) a relative of the finite alternating groups. We find a natural infinite presentation for $V$ as a group generated by these "transpositions," which presentation bears comparison with Dehornoy's infinite presentation and which enables us to develop two small presentations for $V$: a human-interpretable presentation with three generators and eight relations, and a Tietze-derived presentation with two generators and seven relations.