Presentations for the higher dimensional Thompson's groups nV

TL;DR

This paper generalizes Brin's construction of higher dimensional Thompson groups nV, providing finite presentations for all positive integers n and offering an alternative proof of their simplicity.

Contribution

It extends the finite presentation of nV groups to all positive integers n, broadening the understanding of their algebraic structure.

Findings

Finite presentations for all nV groups

Alternative proof of simplicity for nV groups

Generalization of Brin's construction to all n

Abstract

In his papers [2], [3] Brin introduced the higher dimensional Thompson groups nV which are generalizations to the Thompson's group V of self-homeomorphisms of the Cantor set and found a finite set of generators and relations in the case n = 2. We show how to generalize his construction to obtain a finite presentation for every positive integer n. As a corollary, we obtain another proof that the groups nV are simple (first proved by Brin in [4]).