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PERMUTATION-BASED PRESENTATIONS FOR BRIN’S HIGHER-DIMENSIONAL THOMPSON GROUPS
$\boldsymbol {nV}$
- Part of
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- Journal:
- Journal of the Australian Mathematical Society / Volume 116 / Issue 1 / February 2024
- Published online by Cambridge University Press:
- 15 November 2022, pp. 39-67
- Print publication:
- February 2024
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- Article
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The higher-dimensional Thompson groups
$nV$, for 
$n \geqslant 2$, were introduced by Brin [‘Presentations of higher dimensional Thompson groups’, J. Algebra 284 (2005), 520–558]. We provide new presentations for each of these infinite simple groups. The first is an infinite presentation, analogous to the Coxeter presentation for the finite symmetric group, with generating set equal to the set of transpositions in 
$nV$ and reflecting the self-similar structure of n-dimensional Cantor space. We then exploit this infinite presentation to produce further finite presentations that are considerably smaller than those previously known.
