Permutation-based presentations for Brin's higher-dimensional Thompson groups nV

The higher-dimensional Thompson groups nV , for n⩾2 , were introduced by Brin [‘Presentations of higher dimensional Thompson groups’, J. Algebra284 (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.