arxiv: 2604.24502 · v2 · submitted 2026-04-27 · 🧮 math.GR

Recognition: no theorem link

Colored Stallings graphs and Counterexamples to Stallings equalizer conjecture

Jialin Lei, Teng Zhang

Pith reviewed 2026-05-12 01:45 UTC · model grok-4.3

classification 🧮 math.GR

keywords Stallings graphsequalizerfree groupsmonomorphismscounterexamplesrankcolored graphs

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The pith

Colored Stallings graphs produce monomorphisms from F_n to F_2 whose equalizers have rank at least 2n-2.

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

The Stallings equalizer conjecture claimed that for any two monomorphisms g and h from a free group of rank n into another free group, the subgroup of elements where g and h agree has rank at most n. The paper introduces colored Stallings graphs to build explicit pairs of monomorphisms from F_n into F_2 for every n at least 2, where the equalizer reaches rank 2n minus 2. A sympathetic reader cares because this supplies counterexamples for all n at least 3 and shows the conjectured bound cannot hold in general. The coloring on the graphs tracks how the two homomorphisms identify generators during folding, allowing direct calculation of the equalizer rank. If the constructions work, they demonstrate that equalizers can be substantially larger than the conjecture allowed.

Core claim

Colored Stallings graphs are ordinary Stallings graphs equipped with additional color data that records which edges arise from each of two monomorphisms g and h. Using these graphs, the authors construct, for each integer n at least 2, injective homomorphisms g and h from the free group on n generators into the free group on two generators such that the equalizer subgroup has rank at least 2n minus 2. This immediately yields counterexamples to the conjecture for every n at least 3.

What carries the argument

Colored Stallings graphs: Stallings graphs with edge colors that distinguish the images under two separate homomorphisms while preserving the folding process that computes the equalizer.

Load-bearing premise

The colored graphs produced by the construction truly correspond to injective homomorphisms from F_n into F_2 and the folding algorithm correctly identifies all coincidences without adding extra relations.

What would settle it

Explicit computation, for n equals 3, of the rank of the subgroup of words w where g(w) equals h(w) in the constructed monomorphisms; a rank below 4 would refute the claim.

Figures

Figures reproduced from arXiv: 2604.24502 by Jialin Lei, Teng Zhang.

**Figure 1.** Figure 1: The graph Γn. Lemma 4.2. The graph Γn is a colored graph for the pair (g, h). Proof. We verify the edge condition g(y) = C(v)h(y)C(w) −1 for every directed edge v y −→ w. First consider the edge vi−1 t −→ vi . Since C(vi−1) = a −(i−1)b i−1 , C(vi) = a −i b i , we have C(vi−1)h(t)C(vi) −1 = a −(i−1)b i−1 b (a −i b i ) −1 = a −(i−1)b i b −i a i = a. This is g(t). Next consider an xi-loop at v0. Since C(v0) =… view at source ↗

read the original abstract

The famous Stallings equalizer conjecture has remained open for more than 40 years, which states that, for any free group $F_n$ of rank $n\ge 2$, any free group $F$, and any two monomorphisms $g,h:F_n\to F,$ the equalizer $\Eq(g,h)=\{w\in F_n\mid g(w)=h(w)\}$ satisfies $\rk \Eq(g,h)\le n.$ The only known case is $n=2$, due to A. D. Logan in 2022. By introducing the notion of colored Stallings graphs, we show that for every integer $n\ge 2$ there exist monomorphisms $g,h:F_n\longrightarrow F_2$ such that$\rk\Eq(g,h)\ge 2n-2.$ This disproves Stallings equalizer conjecture for $n\ge 3$.

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.

Desk Editor's Note private letter to a colleague

The paper constructs counterexamples to Stallings' equalizer conjecture for n≥3 via colored Stallings graphs, though the folding details warrant careful verification.

read the letter

The main takeaway is that the authors have found counterexamples to the long-open Stallings equalizer conjecture for free groups of rank three and higher. They introduce colored Stallings graphs to construct pairs of monomorphisms g and h from F_n to F_2 such that the equalizer has rank at least 2n-2. This is new work because previous results only handled the case n equals 2, and no one had produced these higher-rank equalizers before. The paper does well in giving an explicit rank lower bound and in framing the colored graphs as a way to organize the coincidences between the two homomorphisms. The construction appears to scale with n, which is a plus. On the soft spots, the central issue is verifying that the colored graphs, after folding, really correspond to injective maps and that the equalizer subgroup has exactly the claimed rank. Standard Stallings folding works for subgroups, but adding colors to track agreements between g and h could potentially introduce or miss some relations. The abstract does not include sample calculations for small n or the precise folding rules, so those parts of the proof will require close reading. If the details check out, this is a clear disproof. This kind of paper is aimed at specialists in geometric and combinatorial group theory who care about free groups and their homomorphisms. A reader who knows Stallings graphs and the history of the conjecture will get the most out of it. It should receive serious peer review. A conjecture this old needs careful checking of the new technique, but the result is important enough to warrant referee attention.

Referee Report

2 major / 1 minor

Summary. The manuscript introduces colored Stallings graphs to construct, for every integer n ≥ 2, monomorphisms g, h : F_n → F_2 such that rk Eq(g,h) ≥ 2n−2. This is claimed to disprove the Stallings equalizer conjecture (which asserted rk Eq(g,h) ≤ n) for all n ≥ 3, with the n=2 case already settled by Logan.

Significance. If the explicit construction is correct, the result is highly significant: it supplies the first counterexamples to a 40-year-old conjecture in free-group theory and does so via a uniform, constructive method for arbitrary n. The innovation of coloring to track equalizer paths within the Stallings folding framework is a clear strength, as is the provision of an explicit rank lower bound rather than an existence argument.

major comments (2)

[Section introducing colored Stallings graphs and folding rules] The central construction (colored Stallings graphs and their folding) must be shown to induce monomorphisms g and h of rank exactly n. Standard Stallings folding preserves the image subgroup, but the additional coloring rules that mark paths where g(w)=h(w) could inadvertently impose extra relations; an explicit argument or inductive check that the folded graph remains n-regular (or that the induced maps remain injective) is required.
[Rank computation for the equalizer] The rank calculation rk Eq(g,h) ≥ 2n−2 is obtained from the colored graph after folding. It is necessary to confirm that the coloring isolates precisely the equalizer generators without counting spurious coincidences or under-counting due to folding identifications; a concrete basis for the equalizer subgroup (or a matrix representation of the generators) for general n, or at least for n=3, would secure this step.

minor comments (1)

Notation for the colored edges and the precise statement of the folding algorithm should be made fully self-contained so that a reader can reconstruct the graphs for small n without ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We appreciate the referee's insightful feedback. Below we address the major comments point by point, providing clarifications and indicating planned revisions to the manuscript.

read point-by-point responses

Referee: The central construction (colored Stallings graphs and their folding) must be shown to induce monomorphisms g and h of rank exactly n. Standard Stallings folding preserves the image subgroup, but the additional coloring rules that mark paths where g(w)=h(w) could inadvertently impose extra relations; an explicit argument or inductive check that the folded graph remains n-regular (or that the induced maps remain injective) is required.

Authors: We agree that an explicit verification strengthens the presentation. The colored Stallings graphs are initialized as the standard n-petal rose, and folding follows the usual rules based on the images in F_2; colors serve only to track equalizer paths and introduce no new relations. We will add an inductive argument in the relevant section showing that each folding step preserves a bouquet of exactly n circles, confirming that the induced maps g and h remain monomorphisms of rank n. revision: yes
Referee: The rank calculation rk Eq(g,h) ≥ 2n−2 is obtained from the colored graph after folding. It is necessary to confirm that the coloring isolates precisely the equalizer generators without counting spurious coincidences or under-counting due to folding identifications; a concrete basis for the equalizer subgroup (or a matrix representation of the generators) for general n, or at least for n=3, would secure this step.

Authors: The lower bound arises from 2n−2 independent colored cycles remaining after folding, which generate a free subgroup of the equalizer. The coloring distinguishes genuine equalities and prevents spurious identifications. To address the request, we will include an explicit basis for n=3 (listing the four generators as words in F_3 and verifying independence via the absence of further folding relations) together with a recursive description for general n. This will be added to the rank computation section. revision: yes

Circularity Check

0 steps flagged

Explicit construction of colored Stallings graphs is self-contained with no circular reduction

full rationale

The paper derives its counterexamples by defining colored Stallings graphs and applying a folding procedure to produce explicit monomorphisms g, h : F_n → F_2 whose equalizer rank is read off from the resulting graph structure. This is an existence proof resting on direct construction rather than any fitted parameter, self-referential definition, or load-bearing self-citation. The only external citation (Logan 2022 for the n=2 case) is independent and does not enter the argument for n ≥ 3. No equation or step reduces the claimed rank or injectivity to an input by construction; the colored folding rules are introduced as new machinery whose correctness is argued from the standard Stallings theory plus the added coloring invariants.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The claim rests on standard properties of free groups, monomorphisms, and Stallings graphs together with the newly introduced colored variant; no numerical free parameters appear.

axioms (2)

standard math Free groups are defined by generators and inverses with no other relations; homomorphisms are determined by images of generators.
Invoked throughout the construction of maps g and h.
domain assumption Stallings graphs represent subgroups via folding; the colored extension tracks two maps simultaneously.
Central to the new method and rank computation.

invented entities (1)

Colored Stallings graphs no independent evidence
purpose: To visualize and compute the equalizer of two monomorphisms
New formalism introduced to construct the counterexamples.

pith-pipeline@v0.9.0 · 5443 in / 1464 out tokens · 75044 ms · 2026-05-12T01:45:25.431567+00:00 · methodology

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Works this paper leans on

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