Non-injectivity of the trace map for character varieties

Arpan Kabiraj; Deblina Das

arxiv: 2605.18649 · v1 · pith:HV3WB7M5new · submitted 2026-05-18 · 🧮 math.GT

Non-injectivity of the trace map for character varieties

Deblina Das , Arpan Kabiraj This is my paper

Pith reviewed 2026-05-20 08:10 UTC · model grok-4.3

classification 🧮 math.GT

keywords character varietiestrace mapGoldman bracketnon-injectivityAmitsur-Levitzki identitysurface groupsfree homotopy classesGL_n representations

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The pith

The Goldman trace map from free homotopy classes of curves to functions on GL_n-character varieties is never injective.

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

The paper proves that the Goldman trace map fails to be injective for any n. It constructs, for each n, an explicit nonzero linear combination of free homotopy classes of closed curves on a closed oriented surface of genus at least two such that the corresponding trace function vanishes on every representation of the fundamental group into GL_n. The argument applies the Amitsur-Levitzki identity to matrices and selects words from a free subgroup of the surface group so that the linear combination stays nonzero at the level of free homotopy classes. This supplies uniform explicit kernel elements and confirms Goldman's earlier prediction of non-injectivity in arbitrary rank.

Core claim

The Goldman trace map is never injective. For each n we construct an explicit nonzero element of the vector space generated by free homotopy classes whose associated trace function vanishes on every homomorphism from π₁(Σ) to GL_n. The construction is based on the Amitsur-Levitzki identity together with a choice of words in a free subgroup of π₁(Σ) ensuring that no cancellation occurs at the level of free homotopy classes.

What carries the argument

Application of the Amitsur-Levitzki identity to a linear combination of words in a free subgroup of the surface fundamental group, chosen to remain nonzero in the space of free homotopy classes.

If this is right

The kernel of the trace map is nontrivial for every positive integer n.
Explicit, uniform families of kernel elements exist for GL_n in every rank.
The map fails to be injective when viewed as a map into the Poisson algebra of regular functions on the character variety.
The same non-injectivity statement holds for reductive linear Lie groups once the GL_n case is established.

Where Pith is reading between the lines

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

The same style of construction may produce kernel elements for SL_n or other reductive groups.
These explicit relations could be used to describe the ideal of functions that vanish on all representations.
For low-genus surfaces the kernel elements might be computable by hand and reveal the structure of the image of the trace map.

Load-bearing premise

The chosen words in the free subgroup produce a linear combination that does not cancel and therefore remains nonzero in the vector space of free homotopy classes.

What would settle it

For a fixed small n and genus-2 surface, compute the explicit linear combination and check whether it is the zero element in the homotopy class space or whether its trace function fails to vanish on some explicit homomorphism to GL_n.

read the original abstract

Given a closed oriented surface $\Sigma$ of genus at least two, the Goldman trace map defines a function from the vector space generated by the free homotopy classes of oriented closed curves to the Poisson algebra of regular functions on the $G$-character variety where $G$ is a reductive (real or complex) linear Lie group. In this article, we prove that this map is never injective. For each $n$, we construct an explicit nonzero element of the vector space whose associated trace function vanishes on every homomorphism from $\pi_1(\Sigma)$ to $GL_n$. The construction is based on the Amitsur-Levitzki identity, together with a choice of words in a free subgroup of $\pi_1(\Sigma)$, ensuring that no cancellation occurs at the level of free homotopy classes. This gives a uniform family of explicit kernel elements, proving Goldman's predicted non-injectivity of the trace map in arbitrary rank.

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

This paper gives an explicit uniform construction of nonzero kernel elements for the Goldman trace map on GL_n character varieties via the Amitsur-Levitzki identity.

read the letter

The main point is that the trace map from free homotopy classes to functions on the character variety is never injective for any n. They build an explicit linear combination of curves whose associated trace function is identically zero on all representations into GL_n, and they show the combination itself is nonzero in the vector space of classes. This directly addresses Goldman's open question with a construction that works uniformly across ranks and for general reductive groups. What they do well is translate the matrix polynomial identity into group words and then embed those words into a free subgroup of the surface group to block conjugacy or inversion cancellations. The vanishing part follows immediately from the identity once the words are fixed, and the approach stays algebraic without extra assumptions. The soft spot is the non-cancellation step. The stress-test note flags that if the specific words coming from Amitsur-Levitzki produce conjugacy-class cancellations for every choice of free subgroup, the element collapses to zero and the kernel claim fails. The paper says the choice avoids this, but that needs checking in the details; it is the only place where the argument could break. No circularity or invented objects appear, and the citations are standard. This is for people working on character varieties, the Goldman bracket, and Poisson structures on representation spaces. A reader who cares about the ring of functions on these varieties will get concrete kernel elements to work with. It deserves a serious referee to verify the word selection and independence claim. I would send it out for review.

Referee Report

1 major / 2 minor

Summary. The paper proves that the Goldman trace map from the vector space generated by free homotopy classes of oriented closed curves on a closed oriented surface Σ of genus at least two to the regular functions on the G-character variety (G reductive) is never injective. For each n it constructs an explicit nonzero element whose associated trace function vanishes on every homomorphism π₁(Σ) → GL_n, using the Amitsur-Levitzki identity together with a choice of words inside a free subgroup of π₁(Σ) chosen so that no cancellation occurs among the free homotopy classes.

Significance. If the non-cancellation step holds, the result supplies the first uniform, explicit family of kernel elements for the trace map in arbitrary rank, confirming Goldman's prediction with a concrete algebraic construction rather than an existence argument. The reliance on the Amitsur-Levitzki identity and standard facts about free subgroups gives a reproducible method that may extend to other Lie groups or Poisson structures on character varieties.

major comments (1)

[Construction paragraph and surrounding discussion] Construction (abstract and main body): the claim that the linear combination remains nonzero in the vector space of free homotopy classes rests on the assertion that the specific words furnished by the Amitsur-Levitzki identity, once embedded into a free subgroup of π₁(Σ), produce no conjugacy-class cancellations. A dedicated lemma or explicit verification for small n (e.g., n=2 or n=3) showing that the resulting classes [γ_i] are linearly independent under the chosen embedding would make this load-bearing step fully transparent.

minor comments (2)

[Introduction] The abstract is concise but the introduction would benefit from a short paragraph recalling the precise definition of the Goldman trace map and the vector space it acts on, for readers outside the immediate area.
Notation for the vector space of free homotopy classes and for the trace functions could be introduced once and used consistently; occasional shifts between [γ] and the curve itself slightly reduce readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for providing constructive feedback that helps clarify the key steps in the argument. We address the major comment below.

read point-by-point responses

Referee: [Construction paragraph and surrounding discussion] Construction (abstract and main body): the claim that the linear combination remains nonzero in the vector space of free homotopy classes rests on the assertion that the specific words furnished by the Amitsur-Levitzki identity, once embedded into a free subgroup of π₁(Σ), produce no conjugacy-class cancellations. A dedicated lemma or explicit verification for small n (e.g., n=2 or n=3) showing that the resulting classes [γ_i] are linearly independent under the chosen embedding would make this load-bearing step fully transparent.

Authors: We agree that the non-cancellation property is a load-bearing step and that making it fully explicit will strengthen the presentation. In the revised version we will add a dedicated lemma (placed immediately after the statement of the main construction) that proves the linear independence of the free homotopy classes. The lemma proceeds by selecting the free subgroup generated by sufficiently high powers of a fixed basis for π₁(Σ) so that all words furnished by the Amitsur-Levitzki identity are cyclically reduced and have disjoint supports in the basis of reduced cyclic words; consequently their linear combination cannot cancel. We will also include explicit verifications for n=2 and n=3, listing the resulting homotopy classes and confirming that the coefficient vector is nonzero in the vector space of free homotopy classes. revision: yes

Circularity Check

0 steps flagged

No circularity; construction uses external identity and standard facts on free groups.

full rationale

The paper's derivation constructs a linear combination of free homotopy classes via the Amitsur-Levitzki identity (an external algebraic theorem) such that the associated trace vanishes identically on all GL_n-representations, then selects words inside a free subgroup of π₁(Σ) to guarantee the combination is nonzero as an element of the vector space. This non-vanishing step relies on the standard fact that distinct reduced words in a free group determine distinct conjugacy classes (hence distinct generators in the homotopy class vector space), without any reduction to a fitted parameter, self-definition, or load-bearing self-citation. The central non-injectivity claim therefore follows from independent external input rather than collapsing to its own assumptions by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The proof depends on two standard external facts: the Amitsur-Levitzki polynomial identity for matrices and the existence of free subgroups inside surface groups of genus at least two. No free parameters or new postulated entities are introduced.

axioms (2)

standard math Amitsur-Levitzki identity: certain noncommutative polynomials vanish identically on n-by-n matrices
Invoked to force the trace function to be zero on all representations into GL_n.
standard math Surface groups of genus >=2 contain free subgroups of rank 2
Used to select words whose free homotopy classes do not cancel.

pith-pipeline@v0.9.0 · 5687 in / 1429 out tokens · 50804 ms · 2026-05-20T08:10:42.562768+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We prove that this map is never injective. For each n, we construct an explicit nonzero element of the vector space whose associated trace function vanishes on every homomorphism from π₁(Σ) to GL_n.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

[1]

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work page 1950
[2]

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David Ginzburg and Ze´ ev Rudnick,Stable multiplicities in the length spectrum of Riemann surfaces, Israel J. Math.104(1998), 129–144. MR 1622287

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Robert D. Horowitz,Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math.25(1972), 635–649. MR 314993

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MR 422434

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Sikora,Character varieties, Trans

Adam S. Sikora,Character varieties, Trans. Amer. Math. Soc.364(2012), no. 10, 5173–5208. MR 2931326 NON-INJECTIVITY OF TRACE MAP 5 Department of Mathematics, Indian Institute of Technology Palakkad Email address:212114002@smail.iitpkd.ac.in Department of Mathematics, Indian Institute of Technology Palakkad Email address:arpaninto@iitpkd.ac.in

work page 2012

[1] [1]

A. S. Amitsur and J. Levitzki,Minimal identities for algebras, Proc. Amer. Math. Soc.1 (1950), 449–463. MR 36751

work page 1950

[2] [2]

Anderson,Variations on a theme of Horowitz, Kleinian groups and hyperbolic 3- manifolds (Warwick, 2001), London Math

James W. Anderson,Variations on a theme of Horowitz, Kleinian groups and hyperbolic 3- manifolds (Warwick, 2001), London Math. Soc. Lecture Note Ser., vol. 299, Cambridge Univ. Press, Cambridge, 2003, pp. 307–341. MR 2044556

work page 2001

[3] [3]

Math.104(1998), 129–144

David Ginzburg and Ze´ ev Rudnick,Stable multiplicities in the length spectrum of Riemann surfaces, Israel J. Math.104(1998), 129–144. MR 1622287

work page 1998

[4] [4]

2, 263–302

William Goldman,Invariant functions on Lie groups and Hamiltonian flows of surface group representations, Inventiones Mathematicae85(1986), no. 2, 263–302

work page 1986

[5] [5]

Goldman,The symplectic nature of fundamental groups of surfaces, Adv

William M. Goldman,The symplectic nature of fundamental groups of surfaces, Adv. in Math. 54(1984), no. 2, 200–225. MR 762512

work page 1984

[6] [6]

Horowitz,Characters of free groups represented in the two-dimensional special linear group, Comm

Robert D. Horowitz,Characters of free groups represented in the two-dimensional special linear group, Comm. Pure Appl. Math.25(1972), 635–649. MR 314993

work page 1972

[7] [7]

Ilya Kapovich, Gilbert Levitt, Paul Schupp, and Vladimir Shpilrain,Translation equivalence in free groups, Trans. Amer. Math. Soc.359(2007), no. 4, 1527–1546. MR 2272138

work page 2007

[8] [8]

Sean Lawton, Larsen Louder, and D. B. McReynolds,Decision problems, complexity, traces, and representations, Groups Geom. Dyn.11(2017), no. 1, 165–188. MR 3641838

work page 2017

[9] [9]

Group Theory9(2006), no

Donghi Lee,Translation equivalent elements in free groups, J. Group Theory9(2006), no. 6, 809–814. MR 2272719

work page 2006

[10] [10]

Leininger,Equivalent curves in surfaces, Geom

Christopher J. Leininger,Equivalent curves in surfaces, Geom. Dedicata102(2003), 151–177. MR 2026843

work page 2003

[11] [11]

MR 422434

Wilhelm Magnus, Abraham Karrass, and Donald Solitar,Combinatorial group theory, revised ed., Dover Publications, Inc., New York, 1976, Presentations of groups in terms of generators and relations. MR 422434

work page 1976

[12] [12]

Sikora,Character varieties, Trans

Adam S. Sikora,Character varieties, Trans. Amer. Math. Soc.364(2012), no. 10, 5173–5208. MR 2931326 NON-INJECTIVITY OF TRACE MAP 5 Department of Mathematics, Indian Institute of Technology Palakkad Email address:212114002@smail.iitpkd.ac.in Department of Mathematics, Indian Institute of Technology Palakkad Email address:arpaninto@iitpkd.ac.in

work page 2012