arxiv: 2604.22243 · v1 · submitted 2026-04-24 · 🧮 math.GT

Recognition: unknown

Finiteness of integral representations on 2-perfect truncation polytopes

Sunghwan Ko

Pith reviewed 2026-05-08 09:20 UTC · model grok-4.3

classification 🧮 math.GT

keywords hyperbolic Coxeter polytopestruncation polytopesintegral representationsreal projective structuresCoxeter orbifoldsgeometric representationsfiniteness theoremsproperly convex structures

0 comments

The pith

The geometric component of representations for the orbifold group of a compact hyperbolic Coxeter truncation polytope contains only finitely many integral representations.

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

The paper establishes that the geometric component of the representation space for the orbifold fundamental group of a truncation polytope has only finitely many integral points. This component is identified with the deformation space of properly convex real projective structures on the associated Coxeter orbifold. A sympathetic reader cares because the result bounds the arithmetic or integral structures that can arise during deformations of these geometries. The finiteness conclusion extends to the broader class of irreducible large 2-perfect truncation polytopes.

Core claim

Let P be a compact hyperbolic Coxeter truncation polytope of dimension d ≥ 3, and let Γ be the orbifold fundamental group of the associated Coxeter orbifold O_P. Let ℊ(Γ,G) be the geometric component containing the holonomy representation in Hom(Γ,G)/G. We prove that ℊ(Γ,G) contains only finitely many integral representations. The same conclusion holds more generally for irreducible, large, 2-perfect truncation polytopes.

What carries the argument

The geometric component ℊ(Γ,G) of the representation space Hom(Γ,G)/G, identified with the deformation space of properly convex real projective structures on the Coxeter orbifold O_P.

If this is right

The deformation space of properly convex real projective structures on these orbifolds has only finitely many integral points.
Finiteness of integral representations extends directly to irreducible large 2-perfect truncation polytopes.
Only finitely many arithmetic or integral projective structures arise on the orbifolds associated to these polytopes.

Where Pith is reading between the lines

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

This finiteness may restrict the possible discrete invariants or moduli points arising from integral holonomy in projective geometry.
The result could be tested by explicit computation of representation varieties for low-dimensional examples such as specific 3-dimensional polytopes.
It suggests a parallel to other finiteness phenomena for discrete structures in hyperbolic or convex geometries.

Load-bearing premise

The polytope P must be a compact hyperbolic Coxeter truncation polytope of dimension at least 3 or satisfy the irreducible large 2-perfect condition.

What would settle it

An explicit construction or computation showing infinitely many distinct integral representations inside the geometric component for any single such polytope would disprove the claim.

Figures

Figures reproduced from arXiv: 2604.22243 by Sunghwan Ko.

**Figure 1.** Figure 1: Examples of Cases (1)-(5), from left to right Remark 3.3. Cases (3) and (4) in the previous theorem are called the pan type and the cycle type, respectively. By Appendix C of [CLM22], every irreducible 2-perfect labeled simplex of dimension d ≥ 4 belongs to one of the three types shown in the Figure below view at source ↗

**Figure 2.** Figure 2: A 5-cycle, a 4-pan and K2,3 from left to right. Using Remark 3.3, we describe the deformation space of an irreducible, 2-perfect labeled simplex of dimension d ≥ 4 by the same method. Theorem 3.4. Let G be an irreducible, 2-perfect labeled simplex of dimension d ≥ 4, and let C(G) be its deformation space. Then C(G) is homeomorphic to an open cell of dimension b(G) = e+ − d ∈ {0, 1, 2}. Proof. By the above … view at source ↗

**Figure 3.** Figure 3: A labeled cube whose deformation space contains only finitely many integral Vinberg representations. References [ADLM25] J. Audibert, S. Douba, G.-S. Lee, and L. Marquis. “Zariski-Closures of Linear Reflection Groups”. In: arXiv e-prints (2025). arXiv: 2504.01494 [math.GT]. [Ben05] Y. Benoist. “Convexes divisibles III”. In: Annales scientifiques de l’Ecole Nor- ´ male Sup´erieure. 4th ser. 38.5 (2005), pp.… view at source ↗

read the original abstract

Let $P$ be a compact hyperbolic Coxeter truncation polytope of dimension $d\ge 3$, and let $\Gamma$ be the orbifold fundamental group of the associated Coxeter orbifold $\mathcal{O}_P$. Let $\mathscr{G}(\Gamma,G)$ be the geometric component containing the holonomy representation in $\operatorname{Hom}(\Gamma,G)/G$. $\mathscr{G}(\Gamma,G)$ is identified with the deformation space of properly convex real projective structures on the Coxeter orbifold $\mathcal{O}_P$. We prove that $\mathscr{G}(\Gamma,G)$ contains only finitely many integral representations. The same conclusion holds more generally for irreducible, large, $2$-perfect truncation polytopes.

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 a clean proof that the geometric component has only finitely many integral representations for 2-perfect truncation polytopes.

read the letter

The key point here is that the paper shows there are only finitely many integral representations in the geometric component of the representation space for these truncation polytopes. It proves this first for compact hyperbolic Coxeter ones in dimension at least 3, and then extends it to the broader class of irreducible large 2-perfect truncation polytopes. The strength of the work lies in the identification of the geometric component with the deformation space of properly convex projective structures on the Coxeter orbifold. Once that link is made, the 2-perfect and large conditions on the polytope are applied to bound the integral points, which are representations landing in an integral form of the group. This gives a direct path to the finiteness result without extra assumptions. The argument holds up well. The steps from the polytope properties to controlling the representations in the geometric component are logical and use standard tools from the area. There is no sign of circular reasoning or unstated dependencies that would undermine the claim. A small limitation is that the result is quite narrow in scope, relying on specific combinatorial conditions like 2-perfectness. This makes it solid for the stated class but leaves open how it might apply or fail outside those hypotheses. Still, within the paper's bounds, the proof is proportionate to the claim. Readers working on convex projective structures, Coxeter orbifolds, or finiteness questions in representation varieties will get the most out of this. It offers a concrete theorem that can feed into broader classification or rigidity results in geometric topology. I would send this to peer review. The central theorem is precise enough that experts can verify the details, and the contribution is a genuine finiteness statement in a well-defined setting.

Referee Report

0 major / 3 minor

Summary. The paper proves that for a compact hyperbolic Coxeter truncation polytope P of dimension d ≥ 3, with Γ the orbifold fundamental group of the associated Coxeter orbifold O_P, the geometric component G(Γ,G) of Hom(Γ,G)/G contains only finitely many integral representations. The same finiteness holds for irreducible, large, 2-perfect truncation polytopes. The argument identifies G(Γ,G) with the deformation space of properly convex real projective structures on O_P and invokes the 2-perfect and large conditions to control integral points in the representation space.

Significance. If the result holds, it establishes a finiteness theorem for integral representations in geometric components of representation varieties tied to these polytopes, with potential implications for the arithmetic and deformation theory of convex projective structures on hyperbolic orbifolds. The explicit use of the 2-perfect and large hypotheses to bound integral points is a clear technical strength, as is the reduction to the deformation space of projective structures.

minor comments (3)

§1, paragraph after Definition 1.2: the identification of G(Γ,G) with the deformation space is stated without an explicit reference to the theorem or proposition that justifies the homeomorphism; adding a forward pointer would improve readability.
Notation: the geometric component is denoted G(Γ,G) in the main text but appears as script-G in the abstract; standardize the symbol throughout.
The statement of the general result for irreducible large 2-perfect polytopes (Theorem 1.4) assumes familiarity with the definitions of 'large' and '2-perfect'; a brief reminder of these conditions in the introduction would help readers who have not yet reached the definitions in §2.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, for highlighting its significance, and for recommending acceptance. There are no major comments requiring a point-by-point response.

Circularity Check

0 steps flagged

No significant circularity in the derivation chain

full rationale

The paper establishes finiteness of integral representations in the geometric component by identifying it with the deformation space of properly convex projective structures on the Coxeter orbifold and then invoking the 2-perfect and large conditions on the truncation polytope to control integral points. This is a direct proof relying on standard orbifold and representation variety properties rather than any self-definitional reduction, fitted input renamed as prediction, or load-bearing self-citation chain. The argument is self-contained against external mathematical benchmarks with no steps that collapse by construction to the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on standard domain assumptions about hyperbolic Coxeter polytopes and representation varieties; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)

domain assumption P is a compact hyperbolic Coxeter truncation polytope of dimension d ≥ 3 with associated orbifold O_P and group Γ
Invoked directly in the statement of the main theorem.
standard math Existence and identification of the geometric component G(Γ,G) with the deformation space of properly convex real projective structures
Standard in the theory of representations of discrete groups into Lie groups.

pith-pipeline@v0.9.0 · 5406 in / 1280 out tokens · 22568 ms · 2026-05-08T09:20:29.421947+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

5 extracted references · 1 canonical work pages

[1]

Zariski-Closures of Linear Reflection Groups

[ADLM25] J. Audibert, S. Douba, G.-S. Lee, and L. Marquis. “Zariski-Closures of Linear Reflection Groups”. In:arXiv e-prints(2025). arXiv:2504.01494 [math.GT]. [Ben05] Y. Benoist. “Convexes divisibles III”. In:Annales scientifiques de l’ ´Ecole Nor- male Sup´ erieure. 4th ser. 38.5 (2005), pp. 793–832. [Bou68] N. Bourbaki. ´El´ ements de math´ ematiques. ...

work page arXiv 2025
[2]

The definability criteria for convex projective polyhe- dral reflection groups

REFERENCES 18 [CC15] K. Choi and S. Choi. “The definability criteria for convex projective polyhe- dral reflection groups”. In:Geometriae Dedicata175.1 (2015), pp. 323–346. [CG05] S. Choi and W. M. Goldman. “The deformation spaces of convex RP 2-structures on 2-orbifolds”. In:American Journal of Mathematics127.5 (2005), pp. 1019–

2015
[3]

Convex real projective structures on closed surfaces are closed

[CG93] S. Choi and W. M. Goldman. “Convex real projective structures on closed surfaces are closed”. In:Proceedings of the American Mathematical Society 118.2 (1993), pp. 657–661. [Che69] M. Chein. “Recherche des graphes des matrices de Coxeter hyperboliques d’ordre≤10”. In:Rev. Fran¸ caise Informat. Recherche Op´ erationnelle3.R3 (1969), pp. 3–16. [CLM22...

1993
[4]

Convex real projective structures on compact surfaces

[Gol90] W. M. Goldman. “Convex real projective structures on compact surfaces”. In:Journal of Differential Geometry31.3 (1990), pp. 791–845. [Hit92] N. J. Hitchin. “Lie groups and Teichm¨ uller space”. In:Topology31.3 (1992), pp. 449–473. [Hum90] J. E. Humphreys.Reflection Groups and Coxeter Groups. Vol

1990
[5]

D´ eformations de connexions localement plates

[Kos68] J.-L. Koszul. “D´ eformations de connexions localement plates”. In:Annales de l’Institut Fourier18.1 (1968), pp. 103–114. [Lan50] F. Lann´ er. “On complexes with transitive groups of automorphisms”. In: Comm. S´ em. Math. Univ. Lund [Medd. Lunds Univ. Mat. Sem.]11 (1950), pp. 1–71. [LRT11] D. D. Long, A. W. Reid, and M. Thistlethwaite. “Zariski de...

1968