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arxiv: 2603.18410 · v3 · pith:BXNPZQOAnew · submitted 2026-03-19 · 🧮 math.GR · math.GT

On subgroups of Brin-Thompson groups nV

Sadayoshi Kojima , Xiaobing Sheng This is my paper

Pith reviewed 2026-05-15 09:11 UTC · model grok-4.3

classification 🧮 math.GR math.GT
keywords Brin-Thompson groupsnVtorsion locally finiteroots of elementsinfinite orderThompson groupssubgroup structurepiecewise linear homeomorphisms
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The pith

The Brin-Thompson group nV is torsion locally finite for all n at least 1, and for n at least 2 it contains infinite-order elements that admit roots of arbitrarily large order.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that every Brin-Thompson group nV is torsion locally finite. This means any finite set of torsion elements generates a finite subgroup, extending a fact previously known only for the original Thompson group V when n equals 1. For n at least 2 the groups additionally contain elements of infinite order that possess roots of every finite order, a feature that does not occur when n equals 1. A sympathetic reader would care because these groups consist of piecewise linear homeomorphisms of n-dimensional cubes, so the results clarify how the algebraic behavior of torsion and roots changes with dimension.

Core claim

We prove that the Brin-Thompson group nV is torsion locally finite for n ≥ 1 which is known only when n = 1, and nV contains elements of infinite order admitting roots with arbitrary large order for n ≥ 2 which is known to not be true for the n = 1 case.

What carries the argument

The Brin-Thompson group nV, the group of piecewise linear homeomorphisms of the n-cube with finitely many pieces, carries the argument by providing the combinatorial setting in which torsion local finiteness and root existence are verified.

Load-bearing premise

The standard combinatorial definition of nV for n greater than 1 preserves the control over torsion elements and root constructions that holds for n equals 1.

What would settle it

An explicit finite collection of torsion elements in some nV whose generated subgroup is infinite would disprove the torsion local finiteness claim.

Figures

Figures reproduced from arXiv: 2603.18410 by Sadayoshi Kojima, Xiaobing Sheng.

Figure 1
Figure 1. Figure 1: Rescaling map ϕ∗ We now define the concept which plays a key role throughout this paper. Definition 2.5 (Dyadic block). A dyadic block B is defined to be a collection of subblocks B = {Bi} such that (1) each Bi is a subblock, (2) Bi ∩ Bj = ∅ if i 6= j, (3) S i Bi = C n , (4) |B| < ∞, where |B| is the cardinality of B which we call the length of B, namely, i ∈ N runs between 1 and |B|. A dyadic subblock is … view at source ↗
Figure 2
Figure 2. Figure 2: X, Y and X ∧ Y Remark 2.1. The operation ∧ is commutative and associative. Remark 2.2. X ∧ Y is a refinement of both X and Y . 2.2. Brin-Thompson group nV . To define Brin-Thompson group nV , we would like to introduce its combinatorial description for pragmatic computations. Let T be the set of all triples (X, Y, σ), where X = {Xi} and Y = {Yj} are dyadic blocks of C n with the same length |X| = |Y | = m,… view at source ↗
Figure 3
Figure 3. Figure 3: The map ¯π [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (X, X, id) and (X′ , X′ , id) We will give a group structure on an appropriate quotient of T in order to say that π¯ induces a homomorphism. To do this, we introduce a crucial terminology where we represent an element in nV by a pair of dyadic blocks with a permutation. Definition 2.9 (Admissibility). A dyadic block X is said to be admissible for g ∈ nV , if g can be represented by a triple (X, Y, σ) for s… view at source ↗
Figure 5
Figure 5. Figure 5: An example of (X, X, σ) Proof. This follows from the fact that g(X) = X and σ is of finite order. The following proposition is motivated by a suggestion from Collin Bleak through private communication with the second author. Proposition 3.3. If g ∈ nV is torsion of order p, then there exists a dyadic block B such that g(B) = B [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: h and its root h1 of order 2 Divide the dyadic block with a single member vertically into two subblocks and squeeze D to the right half in order to obtain D1 and similarly, we divide the single dyadic block vertically into two subblocks and squeeze R to the left half to obtain R1. Giving labelings by combining that of h and a rotation of order 2 in the horizontal direction, we obtain the resulted block pai… view at source ↗
Figure 7
Figure 7. Figure 7: hi−1 and its root hi of order 2 To be more precise, hi here is obtained from hi−1 by squeezing the vertical shifting part into its half horizontally along the central vertical line and filling the rest by vertical subblocks with width 1/2 i evenly. Then, let hi be the element which is generated by a horizontal shift with negative direction together with a vertical shift from the right block of the central … view at source ↗
read the original abstract

We prove that the Brin-Thompson group $nV$ is torsion locally finite for $ n \geq 1$ which is known only when $n = 1$, and $nV$ contains continuum many copies of the additive group of the rationals $\mathbb{Q}$ for $n \geq 2$ which is known to be false for the $n = 1$ case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that the Brin-Thompson group nV is torsion locally finite for every n ≥ 1, extending the known case for n=1, and that for n ≥ 2, nV contains elements of infinite order admitting roots of arbitrarily large finite order, a property that fails for n=1. The arguments rely on explicit combinatorial constructions using the standard piecewise-linear action of nV on the n-colored Cantor set, including verification that finite generating sets produce finite subgroups and that explicit conjugacy and root-taking maps remain inside nV.

Significance. If the results hold, they advance the study of generalized Thompson groups by supplying explicit, combinatorial extensions of known n=1 properties to higher n. The torsion-local-finiteness proof via finite subgroups generated by finite sets and the root-existence construction via verified maps inside nV constitute concrete, falsifiable contributions that could support further work on subgroup structure and dynamics in these groups.

minor comments (2)
  1. [Introduction] The introduction would benefit from a short, self-contained recall of the combinatorial definition of nV (via n-colored Cantor set partitions) to aid readers who know only the n=1 case.
  2. In the root-existence section, the verification that the constructed root maps preserve the nV relations could be expanded with one additional diagram or explicit coordinate check for n=2 to illustrate the general pattern.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our results on torsion local finiteness of nV for all n ≥ 1 and the existence of infinite-order elements with roots of arbitrarily large order for n ≥ 2. The report recommends minor revision but lists no specific major comments. We have no points to address point-by-point and stand ready to incorporate any minor suggestions once provided.

Circularity Check

0 steps flagged

No significant circularity; proofs extend n=1 results via explicit combinatorial constructions

full rationale

The manuscript supplies direct combinatorial arguments based on the standard piecewise-linear action of nV on the Cantor set with n colors. Torsion-local-finiteness is shown by verifying that any finite set of generators produces a finite subgroup, and the root-existence result for infinite-order elements uses explicit conjugacy and root-taking maps that remain inside nV. These steps rely on the known definition and properties for n=1 together with the generalization to higher n, without any self-definitional reductions, fitted inputs renamed as predictions, or load-bearing self-citations that collapse the central claims. The derivation is self-contained against external benchmarks and does not reduce any prediction or uniqueness statement to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper rests on the established definition of Brin-Thompson groups nV and standard facts from geometric group theory; no new free parameters or invented entities are introduced in the abstract.

axioms (1)
  • domain assumption Brin-Thompson groups nV are defined via their standard piecewise-linear or combinatorial action on the n-dimensional Cantor set, as in prior literature.
    The abstract treats the groups as already constructed and extends known n=1 properties.

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