Groups, Geometry, and Dynamics
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Published online: 2017-12-07
The infinite simple group $V$ of Richard J. Thompson: presentations by permutationsCollin Bleak and Martyn Quick (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