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arxiv: 1907.07797 · v1 · pith:KT4UVBQOnew · submitted 2019-07-17 · 🧮 math.GR

One-relator quotients of Partially Commutative Groups

Andrew J. Duncan , Arye Juh\'asz This is my paper

Pith reviewed 2026-05-24 19:41 UTC · model grok-4.3

classification 🧮 math.GR
keywords partially commutative groupsone-relator quotientsMagnus FreiheitssatzMagnus subgroupsembedding theoremsword problemcommutation graphscyclically reduced elements
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The pith

Certain Magnus subgroups embed in one-relator quotients of partially commutative groups under conditions on the relator.

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

The paper generalizes Magnus's Freiheitssatz to partially commutative groups. It establishes that for a relator r meeting the conditions, Magnus subgroups embed into the quotient by the normal closure of r. If r is a proper power s to the n, the root s keeps order n in the quotient. Under slightly stronger conditions the word problem becomes decidable. Readers care because this provides embedding and decidability tools for quotients in groups with partial commutation relations, extending classical results in combinatorial group theory. The authors also reduce the embedding question to minimal parabolic subgroups and show the conditions hold for most elements in certain commutation graphs.

Core claim

We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element r of a partially commutative group ℊ, certain Magnus subgroups embed in the quotient G=ℊ/N(r); that if r=s^n has root s in ℊ then the order of s in G is n, and under slightly stronger conditions that the word problem of G is decidable. We also give conditions under which the question of which Magnus subgroups of ℊ embed in G reduces to the same question in the minimal parabolic subgroup of ℊ containing r.

What carries the argument

The conditions on the relator r that permit the embedding of Magnus subgroups into the one-relator quotient, generalizing the Freiheitssatz.

If this is right

  • Certain Magnus subgroups of the partially commutative group embed into the quotient.
  • If r = s^n then the order of s remains n in the quotient.
  • The word problem is decidable in the quotient under slightly stronger conditions on r.
  • The embedding question reduces to the minimal parabolic subgroup containing r.
  • For commutation graphs that are cycles with a chord, almost all cyclically reduced elements satisfy the conditions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The result opens the way to deciding the word problem in a larger class of one-relator quotients of right-angled Artin groups.
  • Characterizations via the commutation graph may allow explicit lists of embeddable subgroups for specific graphs.
  • Similar techniques could apply to the conjugacy problem in these quotients.
  • Extensions to other classes of groups with commutation relations might follow from the parabolic reduction.

Load-bearing premise

The relator r satisfies the unspecified sufficiently strong conditions that enable the generalization of the Freiheitssatz and the embedding and order preservation statements.

What would settle it

An explicit partially commutative group on a cycle graph with chord, together with a cyclically reduced relator r satisfying the conditions of the theorem, in which some Magnus subgroup fails to embed in the quotient or a root changes its order.

Figures

Figures reproduced from arXiv: 1907.07797 by Andrew J. Duncan, Arye Juh\'asz.

Figure 1
Figure 1. Figure 1: Example 1.7.4 s ′ = a2an−2t ∈ G′ . Then G′ , t and s ′ satisfy the conditions of Theorem 1.2, so setting r ′ = s ′3 the subgroup ⟨a1, a2, . . . , an−2, an−1⟩ of G′ embeds in G′ = G′ /N(r ′). Next let Cn be n-cycle on the right of [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Example 1.7.5 Since we have assumed that supp(s) = A the first of these possibilities, to￾gether with the assumption that lk(t) is a clique, implies that G is a free Abelian group. In this case we defer to standard results for finitely gen￾erated free Abelian groups; and no longer need our Theorems. Moreover, the form s = wt that we have chosen implies that s is not a t-root. (In￾deed, from the definitions… view at source ↗
Figure 3
Figure 3. Figure 3: Cancelling regions: φ(µ) ≡ p, φ(ν1) ≡ φ(ν2) ≡ q region ∆ and boundary label w. As r ′ is cyclically minimal we may replace (3) with (3’) If ∆ is a region of M and ∆ has a boundary cycle p = a1, . . . , an, then the product φ(a1)⋯φ(an) is reduced and equal in F to a cyclic permutation of r or r −1 . As usual we shall restrict to diagrams which do not have pairs of re￾dundant regions of the following sort. L… view at source ↗
Figure 4
Figure 4. Figure 4: Shuffling in the boundary label of ∆. [21, Page 292]. Here we describe modifications sufficient for our particular case. There are two types of these; the first of which results in a new diagram in which the product of edge labels around a boundary cycle may not be freely reduced. The second type may then be used to freely reduce labels on edges of boundary cycles if necessary. Shuffling labels. Suppose th… view at source ↗
Figure 5
Figure 5. Figure 5: Free reduction of the boundary label of ∆ [PITH_FULL_IMAGE:figures/full_fig_p021_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Labelling with φ¯ to define Sep(∆) 25 [PITH_FULL_IMAGE:figures/full_fig_p025_6.png] view at source ↗
read the original abstract

We generalise a key result of one-relator group theory, namely Magnus's Freiheitssatz, to partially commutative groups, under sufficiently strong conditions on the relator. The main theorem shows that under our conditions, on an element $r$ of a partially commutative group $\mathbb{G}$, certain Magnus subgroups embed in the quotient $G=\mathbb{G}/N(r)$; that if $r=s^n$ has root $s$ in $\mathbb{G}$ then the order of $s$ in $G$ is $n$, and under slightly stronger conditions that the word problem of $G$ is decidable. We also give conditions under which the question of which Magnus subgroups of $\mathbb{G}$ embed in $G$ reduces to the same question in the minimal parabolic subgroup of $\mathbb{G}$ containing $r$. In many cases this allows us to characterise Magnus subgroups which embed in $G$, via a condition on $r$ and the commutation graph of $\mathbb{G}$, and to find further examples of quotients $G$ where the word and conjugacy problems are decidable. We give evidence that situations in which our main theorem applies are not uncommon, by proving that for cycle graphs with a chord $\Gamma$, almost all cyclically reduced elements of the partially commutative group $\mathbb{G}(\Gamma)$ satisfy the conditions of the theorem.

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 paper generalizes Magnus's Freiheitssatz to partially commutative groups under sufficiently strong conditions on the relator r. The main theorem shows that certain Magnus subgroups embed in the quotient G=ℊ/N(r), that if r=s^n has root s in ℊ then the order of s in G is n, and under slightly stronger conditions that the word problem of G is decidable. It also gives conditions under which the embedding question reduces to the minimal parabolic subgroup containing r, and proves that for cycle graphs with a chord, almost all cyclically reduced elements satisfy the conditions.

Significance. If the results hold, this extends a classical theorem from one-relator groups to partially commutative groups, with direct implications for embeddings, root orders, and decidability of the word problem. The reduction to minimal parabolic subgroups and the explicit density result for cycle-with-chord graphs supply concrete applications and new examples with decidable word and conjugacy problems. The paper builds directly on the classical Freiheitssatz without circularity.

minor comments (2)
  1. [Abstract] Abstract: the phrase 'sufficiently strong conditions' is used before the main theorem is stated; a parenthetical reference to the precise conditions (as defined in the main theorem) would improve immediate readability.
  2. [§4] The reduction theorem (to the minimal parabolic subgroup) is stated clearly but its proof relies on several intermediate lemmas whose numbering and cross-references could be tightened for easier navigation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending minor revision. The report accurately captures the main contributions: the generalization of Magnus's Freiheitssatz under suitable conditions on the relator, the embedding results for Magnus subgroups, preservation of root orders, and decidability of the word problem, together with the reduction to minimal parabolic subgroups and the density statement for cycle graphs with a chord.

Circularity Check

0 steps flagged

No significant circularity; generalization of classical Freiheitssatz

full rationale

The paper's central result is a conditional generalization of Magnus's classical Freiheitssatz to partially commutative groups, with the main theorem explicitly stating the sufficient conditions on the relator r and reducing the embedding question to the minimal parabolic subgroup. The argument relies on the external classical Freiheitssatz rather than any self-citation chain, fitted parameters renamed as predictions, or self-definitional steps. The conditions are defined upfront in the theorem statement, and the evidence that they hold for almost all cyclically reduced elements on cycle graphs with a chord is a separate combinatorial argument. No load-bearing step reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard results from one-relator group theory plus the existence of 'sufficiently strong conditions' on the relator that are not detailed in the abstract.

axioms (2)
  • standard math Magnus's Freiheitssatz holds in the classical one-relator group setting
    The paper explicitly generalizes this known result.
  • ad hoc to paper The relator r satisfies sufficiently strong conditions (unspecified in abstract)
    This is the load-bearing premise for all embedding, order, and decidability statements.

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Reference graph

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