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

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PresentationForV_GGDfinalVersion.pdf (388.8Kb)
Date
2017
Author
Bleak, Collin
Quick, Martyn
Funder
EPSRC
Grant ID
EP/H011978/1
Keywords
Thompson's groups
Simple groups
Presentations
Generators and relations
Permutations
Transpositions
QA Mathematics
T-NDAS
BDC
R2C
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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.
Citation
Bleak , C & Quick , M 2017 , ' The infinite simple group V of Richard J. Thompson : presentations by permutations ' , Groups, Geometry, and Dynamics , vol. 11 , no. 4 , pp. 1401-1436 . https://doi.org/10.4171/GGD/433
Publication
Groups, Geometry, and Dynamics
Status
Peer reviewed
DOI
https://doi.org/10.4171/GGD/433
ISSN
1661-7207
Type
Journal article
Rights
© 2017, EMS Publishing House. This work has been made available online in accordance with the publisher’s policies. This is the author created, accepted version manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.4171/GGD/433
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  • University of St Andrews Research
URL
http://arxiv.org/abs/1511.02123v1
URI
http://hdl.handle.net/10023/12296

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