Groups, Geometry, and Dynamics

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Volume 11, Issue 4, 2017, pp. 1401–1436
DOI: 10.4171/GGD/433

Published online: 2017-12-07

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

Collin Bleak[1] and Martyn Quick[2]

(1) University of St Andrews, UK
(2) University of St Andrews, UK

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.

Keywords: Thompson’s groups, simple groups, presentations, generators and relations, permutations, transpositions

Bleak Collin, Quick Martyn: The infinite simple group $V$ of Richard J. Thompson: presentations by permutations. Groups Geom. Dyn. 11 (2017), 1401-1436. doi: 10.4171/GGD/433