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.

We investigate the notion of relatively amenable topological action and show that the action of Thompson’s group T on $S^1$ is relatively amenable with respect to Thompson’s group F. We use this to conclude that F is exact if and only if T is exact. Moreover, we prove that the groupoid of germs of the action of T on $S^1$ is Borel amenable.

We show that R.J. Thompson’s groups F and T are bi-interpretable with the ring of the integers. From a result by A. Khélif, these groups are quasi-finitely axiomatizable and prime. So, the group T provides an example of a simple group which is quasi-finitely axiomatizable and prime. This answers questions posed by T. Altınel and A. Muranov in [2], and by A. Nies in [12].