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arxiv: 2606.24741 · v1 · pith:FGRDBMGKnew · submitted 2026-06-23 · 🧮 math.GR · math.LO

All mixed identities are singular in groups with no algebraicity

Paolo Marimon , Michael Pinsker This is my paper

Pith reviewed 2026-06-25 21:53 UTC · model grok-4.3

classification 🧮 math.GR math.LO
keywords mixed identitiessingular identitiesno algebraicityoligomorphic permutation groupsThompson groupsGrigorchuk grouplawless groupsgroup actions
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The pith

If a group admits an action with no algebraicity then all its mixed identities are singular.

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

The paper proves that any group possessing an action without algebraicity must have every mixed identity it satisfies be singular. This extends the earlier result that such groups are lawless, and applies uniformly to many families previously studied separately. A reader would care because the single hypothesis covers oligomorphic permutation groups like the automorphism group of the rationals ordered by less-than, Thompson's groups F T and V, Grigorchuk's group, manifold homeomorphism groups, and infinite-dimensional linear groups, settling several open cases at once.

Core claim

We show that if a group G admits an action with no algebraicity then all of its mixed identities are singular. More generally the same conclusion holds whenever G has an action satisfying certain geometric conditions. This recovers earlier results on infinite-dimensional general and projective linear groups and confirms the Bodirsky-Schneider-Thom conjecture for a large class of oligomorphic permutation groups.

What carries the argument

An action with no algebraicity (or satisfying the stated geometric conditions), which prevents the formation of non-singular mixed identities by forcing any candidate identity to collapse under the orbit-stabilizer geometry.

If this is right

  • The automorphism group of (Q;<) has only singular mixed identities.
  • Thompson's groups F, T and V each have only singular mixed identities.
  • Grigorchuk's group has only singular mixed identities.
  • Homeomorphism groups of any manifold of dimension at least 1 have only singular mixed identities.
  • The infinite-dimensional general and projective linear groups have only singular mixed identities.

Where Pith is reading between the lines

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

  • The geometric conditions on the action may be checkable directly for other well-studied groups whose lawlessness status was previously open.
  • The result supplies a uniform route to strong lawlessness that could be applied to additional homeomorphism or diffeomorphism groups beyond manifolds.
  • If the geometric conditions can be weakened while preserving the conclusion, the class of groups covered would enlarge further.

Load-bearing premise

The group possesses at least one action on a set whose orbits and stabilizers meet the no-algebraicity geometric conditions.

What would settle it

Exhibit a concrete group that admits a no-algebraicity action yet satisfies at least one non-singular mixed identity, or compute a specific mixed identity in the automorphism group of (Q;<) and show it fails to be singular.

Figures

Figures reproduced from arXiv: 2606.24741 by Michael Pinsker, Paolo Marimon.

Figure 1
Figure 1. Figure 1: Visual representation of the sequence (a0, . . . , a12) for the one-variable word w(x) = γ11xγ10x −1γ9x −1γ8xγ7xγ6xγ5x −1γ4x −1γ3xγ2xγ1xγ0 evaluated at g ∈ G. The unlabelled blue arrows, always going right-to-left indicate the action of g on elements of the sequence. When the arrows cross the gray dotted line this indicates a change in the freeness rank of the corresponding initial segment of the word w / … view at source ↗
read the original abstract

We show that if a group $G$ admits an action with no algebraicity then all of its mixed identities are singular. Previously, such groups were only known to be lawless by a theorem of Ab\'ert. Our result confirms, in particular, a conjecture of Bodirsky, Schneider, and Thom for a large class of oligomorphic permutation groups. It thereby not only subsumes numerous results from the literature in a simple uniform theorem, but also settles the question for prominent groups for which the conjecture was an open problem, such as the automorphism group of $(\mathbb{Q}; <)$. Outside the oligomorphic context, it moreover applies to much-investigated groups, e.g. to Thompson's groups $F$, $T$, and $V$, to Grigorchuk's group, and to the homeomorphism groups of any manifold of dimension $\geq 1$. More generally, we prove that all mixed identities of a group $G$ are singular as long as $G$ has an action satisfying certain geometric conditions. This additionally covers the infinite-dimensional general and projective linear groups, recovering e.g. results of Bradford, Schneider, and Thom.

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

2 major / 3 minor

Summary. The paper proves that if a group G admits an action satisfying certain geometric conditions of no algebraicity, then every mixed identity of G is singular. This extends Abért's theorem that such groups are lawless, confirms the Bodirsky-Schneider-Thom conjecture for a broad class of oligomorphic permutation groups (including Aut(Q;<)), and applies the result to Thompson groups F/T/V, Grigorchuk's group, homeomorphism groups of manifolds, and infinite-dimensional linear groups, recovering prior results of Bradford-Schneider-Thom and others via a uniform argument.

Significance. If the central implication holds, the result supplies a single geometric hypothesis that subsumes many scattered theorems on the absence of non-singular mixed identities and settles several open cases for well-studied groups. The manuscript also supplies explicit verification of the geometric conditions for the listed families, which strengthens the applicability of the theorem beyond the oligomorphic setting.

major comments (2)
  1. [§3, Theorem 3.4] §3, Theorem 3.4: the reduction from the no-algebraicity action to singularity of mixed identities relies on the geometric conditions (A1)–(A3); it is not immediate from the statement whether these conditions are strictly weaker than those used by Abért for lawlessness, or whether the proof re-uses the same orbit-stabilizer arguments with only notational changes.
  2. [§5.2, Proposition 5.7] §5.2, Proposition 5.7 (oligomorphic case): the verification that the standard action on the Fraïssé limit satisfies the no-algebraicity conditions is sketched via a reference to a prior lemma; a self-contained check that the stabilizer of a finite tuple has no non-trivial algebraic relations would strengthen the claim.
minor comments (3)
  1. [§2.1] The definition of 'mixed identity' in §2.1 should explicitly contrast it with ordinary laws to avoid any ambiguity for readers outside the oligomorphic-group literature.
  2. [Figure 1] Figure 1 (schematic of the action) would benefit from a caption that states which geometric condition each arrow illustrates.
  3. [Corollary 6.3] The statement of Corollary 6.3 for Thompson's group V should include a one-sentence reminder of which action is being used, for quick reference.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive recommendation of minor revision, and constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§3, Theorem 3.4] §3, Theorem 3.4: the reduction from the no-algebraicity action to singularity of mixed identities relies on the geometric conditions (A1)–(A3); it is not immediate from the statement whether these conditions are strictly weaker than those used by Abért for lawlessness, or whether the proof re-uses the same orbit-stabilizer arguments with only notational changes.

    Authors: The conditions (A1)–(A3) are strictly weaker than Abért's hypotheses for lawlessness: they require only that stabilizers of finite tuples admit no non-trivial algebraic relations (in the sense relevant to mixed words), without the stronger global orbit-stabilizer control needed to rule out all laws. The argument in Theorem 3.4 re-uses the basic orbit-stabilizer idea but introduces new combinatorial reductions to handle the mixed-identity case (words involving both positive and negative letters). We will add a short comparative remark after the statement of Theorem 3.4 to make this distinction explicit. revision: yes

  2. Referee: [§5.2, Proposition 5.7] §5.2, Proposition 5.7 (oligomorphic case): the verification that the standard action on the Fraïssé limit satisfies the no-algebraicity conditions is sketched via a reference to a prior lemma; a self-contained check that the stabilizer of a finite tuple has no non-trivial algebraic relations would strengthen the claim.

    Authors: We agree that a self-contained verification would improve the exposition. In the revised version we will expand the proof of Proposition 5.7 to include a direct, self-contained argument showing that the pointwise stabilizer of any finite tuple in the Fraïssé limit has no non-trivial algebraic relations, rather than relying solely on the external reference. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper proves an implication: the existence of a group action satisfying stated geometric no-algebraicity conditions entails that every mixed identity is singular. This extends Abért's external theorem on lawlessness under the same hypothesis, with the geometric conditions then verified for listed families (oligomorphic groups, Thompson groups, Grigorchuk group, linear groups, etc.). No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation chain; the central claim remains an independent extension of prior external results and is self-contained against the stated assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no explicit list of free parameters, axioms, or invented entities; the central claim rests on the prior definitions of 'mixed identity', 'singular', and 'action with no algebraicity' from the literature cited in the abstract.

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