A family of non-isomorphism results

We calculate the local groups of germs associated with the higher dimensional R. Thompson groups nV. For a given n ∈ N ∪ {ω}, these groups of germs are free abelian groups of rank r, for r ≤ n (there are some groups of germs associated with nV with rank precisely k for each index 1 ≤ k ≤ n). By Rubin’s theorem, any conjectured isomorphism between higher dimensional R. Thompson groups induces an isomorphism between associated groups of germs. Thus, if m = n the groups mV and nV cannot be isomorphic. This answers a question of Brin.