  The higher-dimensional Thompson groups $nV$, for $n \geq 2$, were introduced
by Brin in 2005. 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.
