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arxiv: 2409.09630 · v2 · submitted 2024-09-15 · 🧮 math.GR · math.GT· math.PR

Asymptotic Burnside laws

Gil Goffer , Be'eri Greenfeld , Alexander Yu. Olshanskii This is my paper

Pith reviewed 2026-05-23 20:50 UTC · model grok-4.3

classification 🧮 math.GR math.GTmath.PR
keywords Burnside lawsfinitely generated groupsrandom walks on groupsCayley graphsfree subgroupsasymptotic probabilityprobabilistic group theory
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The pith

Finitely generated groups can satisfy a Burnside law with probability 1 while containing free subgroups.

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

The paper constructs finitely generated groups in which the probability that random elements satisfy a Burnside law of the form x^n=1 approaches 1 under uniform measures on balls in the Cayley graph and under every lazy non-degenerate random walk. These groups nonetheless contain free subgroups. The work shows that the limiting probability depends on the choice of generating set for the same group, that two coprime Burnside laws can hold simultaneously with probability 1, and that every real number in [0,1] can appear as a partial limit of the probability sequence.

Core claim

We construct a finitely generated group that satisfies a Burnside law, namely a law of the form x^n=1, with limit probability 1 with respect to uniform measures on balls in its Cayley graph and under every lazy non-degenerate random walk, while containing a free subgroup. We show that the limit probability of satisfying a Burnside law is highly sensitive to the choice of generating set, by providing a group for which this probability is 0 for one generating set and 1 for another. Furthermore, we construct groups that satisfy Burnside laws of two co-prime exponents with probability 1. Finally, we present a finitely generated group for which every real number in the interval [0,1] appears as a

What carries the argument

Geometric analysis of relations in groups together with information-theoretic coding to control the asymptotic densities of words satisfying the given law.

Load-bearing premise

The geometric and combinatorial constructions can be carried out to produce the desired groups without the relations eliminating the free subgroups or other known properties of finitely generated groups.

What would settle it

An explicit computation or alternative construction showing that the probability of satisfying x^n=1 cannot approach 1 in any finitely generated group that contains a free subgroup.

Figures

Figures reproduced from arXiv: 2409.09630 by Alexander Yu. Olshanskii, Be'eri Greenfeld, Gil Goffer.

Figure 1
Figure 1. Figure 1: Convention on orientation More details on Burnside-type Presentations are found in [28, Chapters 7,8]. 3. Geometric preliminaries 3.1. Words. Let S = {x1, . . . , xm} be a finite set. We say that X is a word over S if it is a concatenation of letter from S ±1 = {x ±1 1 , . . . , x±1 m }. Let X and Y be words over a set S. We write X ≡ Y if X and Y consist of the same letters in the same order. A word Y is … view at source ↗
Figure 2
Figure 2. Figure 2: A contiguity subdiagram Lemma 4.1 (Lemma 15.3 in [28]). Let ∆ be a reduced diagram, Γ a contiguity subdiagram of a cell Π to a section q of ∂∆. As in [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Case 1 [PITH_FULL_IMAGE:figures/full_fig_p023_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Case 2 Proof. If ∆ has rank 0, then the edges of the geodesic path z connecting the vertex p− = q− to p+ = q+ have to be among the edges of p and of q, and so cl(z) ≤ ∣z∣ ≤ (p, q) and the statement holds. Therefore ∆ has a positive rank. We now induct on the number of cells in ∆, assuming it is positive. By Lemma 4.3, ∆ has a γ-cell Π. We consider two cases. Case 1. There is a maximal contiguity diagram of… view at source ↗
Figure 5
Figure 5. Figure 5: Case 1 Denote by p1q1p2q2 the boundary path of Γa, by s1t1s2t2the boundary path of Γb, and by q1wt1u the boundary path of Π. Denote by C ′ the vertex (p2)+ (i.e. the end of p2) and by D the vertex (t2)+. The length of the boundary subpaths BC′ , C ′C, and CD are denoted by a ′ , f, and e resp. Let AC′ be a geodesic connecting the vertices A and C ′ . We have that ∣s1w −1 p2∣ < (γ + 2ζ)∣∂Π∣ < 2γ∣∂Π∣, and it… view at source ↗
Figure 6
Figure 6. Figure 6: Case 2 Case 2. Assume that there are only two maximal contiguity subdiagrams of Π, say Γb and Γc to the sides CA and AB, resp. So as in Case 1, each of the contiguity degrees is greater than 1 2 − 3β − γ. Denote by p1q1p2q2 the boundary path of Γc, by s1t1s2t2 the boundary path of Γb, and by q1wt1u the boundary path of Π. Denote further by A ′ (respectively, by D) the vertex (p1)− (resp., the vertex (s2)+)… view at source ↗
Figure 7
Figure 7. Figure 7: Case 3 Case 3. There are three maximal contiguity subdiagrams of Π: Γa, Γb, and Γc to the sides BC, CA and AB, respectively. We align the notations for the boundaries of Γa and Γb with those from Case 2. We now denote B ′ = (p2)+, C′ = (s1)−, v = ∣BB′ ∣, v′ = ∣B ′A∣, z = ∣CC′ ∣, z′ = ∣C ′A∣, whence v + v ′ = c and z + z ′ = b. Assume without loss of generality that v ′ ≤ z ′ . Since BC is λ-geodesic, Π is … view at source ↗
Figure 8
Figure 8. Figure 8: The diagrams ∆ (left) and ∆′ (right). Recall that a word U is called λ-geodesic for some λ ≥ 1 if for every word U ′ which is equal to U in G, we have ∣U∣ ≤ λ∣U ′ ∣. A word U is called geodesic (or minimal) if it is λ-geodesic with λ = 1. Finally, a word U is called cyclically λ-geodesic if every cyclic permutation of U is λ-geodesic. Lemma 8.6. Let ∆ be a reduced annular diagram over G with contours p and… view at source ↗
read the original abstract

We construct novel examples of finitely generated groups that exhibit seemingly-contradicting probabilistic behaviors with respect to Burnside laws. We construct a finitely generated group that satisfies a Burnside law, namely a law of the form $x^n=1$, with limit probability 1 with respect to uniform measures on balls in its Cayley graph and under every lazy non-degenerate random walk, while containing a free subgroup. We show that the limit probability of satisfying a Burnside law is highly sensitive to the choice of generating set, by providing a group for which this probability is $0$ for one generating set and $1$ for another. Furthermore, we construct groups that satisfy Burnside laws of two co-prime exponents with probability 1. Finally, we present a finitely generated group for which every real number in the interval $[0,1]$ appears as a partial limit of the probability sequence of Burnside law satisfaction, both for uniform measures on Cayley balls and for random walks. Our results resolve several open questions posed by Amir, Blachar, Gerasimova, and Kozma. The techniques employed in this work draw upon geometric analysis of relations in groups, information-theoretic coding theory on groups, and combinatorial and probabilistic methods.

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 / 1 minor

Summary. The paper constructs finitely generated groups satisfying a Burnside law x^n=1 with asymptotic probability 1 under uniform measures on Cayley balls and every lazy non-degenerate random walk, while containing a free subgroup. It further exhibits a group where this probability is 0 for one generating set and 1 for another, groups satisfying Burnside laws for two coprime exponents with probability 1, and a group in which every value in [0,1] arises as a partial limit of the probability sequence (for both uniform ball measures and random walks). The results resolve open questions of Amir, Blachar, Gerasimova, and Kozma via geometric analysis of relations, information-theoretic coding on groups, and combinatorial/probabilistic methods.

Significance. If the constructions are valid, the work is significant: it supplies explicit examples separating probabilistic satisfaction of laws from the absence of free subgroups (free factors can remain exponentially sparse), demonstrates sensitivity to generating sets, and realizes arbitrary limit behaviors. These resolve multiple open questions in asymptotic and probabilistic group theory and illustrate the utility of coding-theoretic control of densities.

minor comments (1)
  1. [Abstract] The abstract uses the hyphenated form 'seemingly-contradicting'; standard mathematical English prefers 'seemingly contradictory'.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the paper, recognition of its significance in resolving the open questions of Amir, Blachar, Gerasimova, and Kozma, and recommendation to accept. The report correctly captures the main results on asymptotic probabilities of Burnside laws under ball measures and random walks.

Circularity Check

0 steps flagged

No significant circularity; existence result via independent constructions

full rationale

The paper presents an existence result for finitely generated groups exhibiting specific asymptotic densities for Burnside laws while containing free subgroups. The abstract and description frame this as achieved through geometric analysis of relations, information-theoretic coding, and combinatorial/probabilistic methods, without any equations, fitted parameters, or reductions to self-citations that would make the central claims tautological. No load-bearing steps reduce by construction to inputs; the constructions are described as resolving external open questions from unrelated authors. This is a standard non-circular existence proof in group theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract provides no explicit free parameters, axioms beyond standard group theory, or invented entities; all claims rest on existence of constructions whose details are not supplied.

axioms (1)
  • standard math Standard axioms of group theory and basic properties of Cayley graphs and random walks on groups
    Invoked implicitly when discussing uniform measures on balls and lazy random walks.

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