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.
Collin Bleak; Martyn Quick. The infinite simple group $V$ of Richard J. Thompson: presentations by permutations. Groups, geometry, and dynamics, Tome 11 (2017) no. 4, pp. 1401-1436. doi: 10.4171/ggd/433
@article{10_4171_ggd_433,
author = {Collin Bleak and Martyn Quick},
title = {The infinite simple group $V$ of {Richard} {J.} {Thompson:} presentations by permutations},
journal = {Groups, geometry, and dynamics},
pages = {1401--1436},
year = {2017},
volume = {11},
number = {4},
doi = {10.4171/ggd/433},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/433/}
}
TY - JOUR
AU - Collin Bleak
AU - Martyn Quick
TI - The infinite simple group $V$ of Richard J. Thompson: presentations by permutations
JO - Groups, geometry, and dynamics
PY - 2017
SP - 1401
EP - 1436
VL - 11
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ggd/433/
DO - 10.4171/ggd/433
ID - 10_4171_ggd_433
ER -
%0 Journal Article
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%T The infinite simple group $V$ of Richard J. Thompson: presentations by permutations
%J Groups, geometry, and dynamics
%D 2017
%P 1401-1436
%V 11
%N 4
%U http://geodesic.mathdoc.fr/articles/10.4171/ggd/433/
%R 10.4171/ggd/433
%F 10_4171_ggd_433
