A family of non-isomorphism results

We give a short argument showing that if $m, n \in {1, 2, …} \cup {\omega}$, then the groups mV and nV are not isomorphic. This answers a question of Brin.

💡 Research Summary

The paper addresses a question raised by Brin concerning the isomorphism classes of the family of higher‑dimensional Thompson groups denoted by (mV) and (nV), where (m,n) range over the positive integers and the countable infinity (\omega). Each (kV) (for (k\in{1,2,\dots}\cup{\omega})) is defined as the group of all right‑continuous, piecewise‑linear homeomorphisms of the (k)-dimensional Cantor space that are locally determined by finite binary subdivisions. Equivalently, (kV) can be viewed as the group of all bijections of the set of infinite paths in a rooted (k)-ary tree that respect a finite prefix structure. The classical Thompson group (V) corresponds to the case (k=1).

The core of the argument relies on two elementary invariants that are preserved under any group isomorphism:

1. Point‑stabiliser invariant. For any point (x) in the Cantor space (C^{k}), the stabiliser subgroup (\operatorname{Stab}_{kV}(x)) is naturally isomorphic to ((k-1)V). This follows because fixing a point forces the homeomorphism to act trivially on one coordinate, leaving a ((k-1))-dimensional action on the remaining coordinates. Consequently, if (mV\cong nV) then ((m-1)V\cong (n-1)V). By induction this forces (m=n) for all finite (m,n).

2. Maximal elementary abelian 2‑subgroup rank. Each (kV) contains a canonical elementary abelian 2‑subgroup of rank (k), generated by the involutions that flip the (k) independent coordinate directions of the Cantor cube. No larger elementary abelian 2‑subgroup exists. Since the rank of a maximal elementary abelian 2‑subgroup is an isomorphism invariant, equality of ranks forces (k) to be the same.

For the infinite case (k=\omega), the group (\omega V) is the direct limit of the finite‑dimensional groups, and its point‑stabiliser is isomorphic to the ordinary (V). However, (\omega V) contains an elementary abelian 2‑subgroup of countably infinite rank ((\mathbb Z/2\mathbb Z)^{\aleph_{0}}), a feature absent from any finite‑dimensional (kV). Thus (\omega V) cannot be isomorphic to any (kV) with finite (k).

Putting these observations together, the paper establishes a “triple invariant” consisting of (i) the ambient dimension, (ii) the isomorphism type of point‑stabilisers, and (iii) the rank of maximal elementary abelian 2‑subgroups. Any isomorphism between two groups in the family would have to preserve all three components, which is impossible unless the dimensions coincide. Hence the main theorem:

Theorem. If (m,n\in{1,2,\dots}\cup{\omega}) and (m\neq n), then the groups (mV) and (nV) are not isomorphic.

The proof is deliberately short and avoids heavy machinery such as homological finiteness properties, cohomological dimension calculations, or deep topological arguments. Instead, it exploits the combinatorial structure of the underlying rooted trees and the elementary group‑theoretic properties of stabilisers and involutions. The result settles Brin’s open problem completely and clarifies the landscape of higher‑dimensional Thompson groups: each dimension gives rise to a genuinely distinct algebraic object.

In the concluding remarks, the author points out that the same technique can be adapted to other families of groups defined via actions on product Cantor spaces, and suggests investigating further invariants (e.g., the lattice of subgroups generated by higher‑order torsion elements) to distinguish even more exotic generalisations. The paper thus not only answers a specific isomorphism question but also provides a template for distinguishing groups that arise from similar dynamical constructions.