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arxiv: 2604.09014 · v1 · submitted 2026-04-10 · 🧮 math.GR · math.GT· math.SP

Spectral Dehn functions and a characterisation of word-hyperbolicity

Mayukh Mukherjee This is my paper

Pith reviewed 2026-05-10 16:45 UTC · model grok-4.3

classification 🧮 math.GR math.GTmath.SP
keywords word-hyperbolicityDehn functionspectral Dehn functionvan Kampen diagramsrandom-walk Laplacianfilling lengthquasi-isometry invariantfinitely presented groups
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The pith

A finitely presented group is word-hyperbolic precisely when the infimum of its degree-free spectral Dehn function is positive.

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

The paper defines the spectral Dehn function as the infimum of the first Dirichlet eigenvalue of the random-walk Laplacian taken over all area-minimizing van Kampen diagrams whose boundary length is at most n. It proves a spectral-isoperimetric inequality that relates this quantity to the ordinary Dehn function and shows that a degree-free, face-dual variant has positive infimum if and only if the group is word-hyperbolic. The same construction yields a spectral filling profile, obtained by passing to free completions under a hereditary quasi-minimality condition, that is a quasi-isometry invariant and separates presentations inside the linear Dehn class. A diagramwise inequality FL_b(Delta) times Area(Delta) at least c over lambda_1(Delta) recovers the sharp quadratic exponent 1/2 via a discrete Faber-Krahn inequality.

Core claim

The central claim is that a finitely presented group is word-hyperbolic if and only if the infimum over n of the degree-free spectral Dehn function Lambda star P of n is strictly positive. This equivalence follows from a spectral-isoperimetric inequality together with a diagramwise filling-length bound and the construction of a spectral filling profile via free completions that preserves the positivity criterion under quasi-isometries.

What carries the argument

The spectral Dehn function Lambda_P(n), the infimum of the first Dirichlet eigenvalue of the random-walk Laplacian on area-minimizing van Kampen diagrams of boundary length at most n; its degree-free face-dual variant Lambda star P carries the hyperbolicity characterization.

If this is right

  • Every disk diagram satisfies the filling-length bound FL_b(Delta) times Area(Delta) at least c over lambda_1(Delta).
  • In the quadratic case the exponent 1/2 is sharp, attained by rectangular commutator grids over Z^2.
  • The spectral filling profile obtained from free completions is a quasi-isometry invariant of the group.
  • The profile distinguishes presentations of groups that share the same linear Dehn function.

Where Pith is reading between the lines

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

  • Estimating the first eigenvalue on a finite collection of candidate diagrams could give a practical test for hyperbolicity that bypasses direct search for isoperimetric inequalities.
  • The same spectral construction might extend to other coarse invariants by replacing the Dehn function with other filling profiles.
  • The speed at which the infimum approaches its positive value could supply a numerical measure of hyperbolicity strength inside the hyperbolic class.

Load-bearing premise

Area-minimizing van Kampen diagrams exist for every boundary length and the random-walk Laplacian on them is well-defined and obeys the stated spectral-isoperimetric inequality.

What would settle it

Exhibit a finitely presented group that is word-hyperbolic yet has infimum of Lambda star P equal to zero, or a non-hyperbolic group for which the same infimum is positive.

Figures

Figures reproduced from arXiv: 2604.09014 by Mayukh Mukherjee.

Figure 1
Figure 1. Figure 1: FIGURE 1 [PITH_FULL_IMAGE:figures/full_fig_p044_1.png] view at source ↗
read the original abstract

We introduce a \emph{spectral Dehn function} \[ \Lambda_{\mathcal{P}}(n):=\inf \lambda_1(\Delta), \] where $\lambda_1(\Delta)$ is the first Dirichlet eigenvalue of the random-walk Laplacian on a van Kampen diagram $\Delta$, and the infimum runs over area-minimising diagrams with boundary length at most $n$. We prove a spectral-isoperimetric inequality relating $\Lambda_{\mathcal{P}}$ to the Dehn function, and show that its degree-free face-dual variant $\Lambda^\ast_{\mathcal P}$ characterises word-hyperbolicity: a finitely presented group is word-hyperbolic if and only if \[ \inf_n \Lambda^\ast_{\mathcal{P}}(n)>0. \] Every disk diagram satisfies a diagramwise filling-length bound \[ \mathrm{FL}_b(\Delta)\cdot \operatorname{Area}(\Delta) \ge c/\lambda_1(\Delta); \] combined with a discrete Faber-Krahn inequality, this yields the sharp exponent $1/2$ in the quadratic case, attained by rectangular commutator grids over $\mathbb Z^2$. By passing to the free completion and introducing a hole-free-ancestor hereditary quasi-minimality condition, we obtain a spectral filling profile whose positivity criterion is a quasi-isometry invariant of finitely presented groups and again characterises word-hyperbolicity. The resulting profile carries finer information than the Dehn function: it separates presentations within the linear Dehn class.

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 / 3 minor

Summary. The paper introduces the spectral Dehn function Λ_P(n) as the infimum of the first Dirichlet eigenvalue λ_1(Δ) of the random-walk Laplacian on area-minimizing van Kampen diagrams with boundary length at most n. It proves a spectral-isoperimetric inequality FL_b(Δ) · Area(Δ) ≥ c / λ_1(Δ) for disk diagrams, shows that the degree-free face-dual variant Λ^*_P characterizes word-hyperbolicity (inf_n Λ^*_P(n) > 0 iff the finitely presented group is word-hyperbolic), derives a diagram-wise filling-length bound, and constructs a quasi-isometry invariant spectral filling profile (via free completion and hole-free-ancestor hereditary quasi-minimality) that also characterizes hyperbolicity while separating presentations inside the linear Dehn class.

Significance. If the results hold, the work supplies a new spectral characterization of word-hyperbolicity that is quasi-isometry invariant and strictly finer than the Dehn function, as it distinguishes presentations with identical linear Dehn functions. The diagram-wise bound combined with the discrete Faber-Krahn inequality recovers the sharp 1/2 exponent in the quadratic case (attained by rectangular grids over Z^2). Strengths include the use of standard area-minimization to guarantee existence of diagrams, the well-definedness of the bounded-degree dual graphs, and the explicit passage from linear isoperimetric inequality to uniform Cheeger constant on those graphs.

minor comments (3)
  1. The abstract and introduction introduce Λ^*_P(n) as the 'degree-free face-dual variant' without an immediate inline definition or pointer to the precise modification of the Laplacian; adding one sentence clarifying the degree-free construction would improve readability for readers outside spectral graph theory.
  2. The quasi-isometry invariance argument relies on the free completion and the hole-free-ancestor hereditary quasi-minimality condition; a short illustrative example (even a small diagram with a hole) would make the hereditary property easier to verify.
  3. Notation for the filling-length bound FL_b(Δ) is used before its definition; a forward reference or brief reminder in the statement of the spectral-isoperimetric inequality would help.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their careful summary of the manuscript and for their positive assessment of its contributions. We appreciate the recognition that the spectral Dehn function and its variants provide a quasi-isometry invariant characterization of word-hyperbolicity that is strictly finer than the Dehn function, along with the diagram-wise bounds and connections to the discrete Faber-Krahn inequality. The recommendation for minor revision is noted, and we will prepare a revised version accordingly. No specific major comments were raised.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained from independent definitions and inequalities

full rationale

The paper defines Λ_P(n) directly as the infimum of λ1(Δ) over area-minimizing van Kampen diagrams with boundary length ≤ n, without reference to hyperbolicity. It establishes the spectral-isoperimetric inequality FL_b(Δ) · Area(Δ) ≥ c/λ1(Δ) from diagram properties, invokes the standard discrete Faber-Krahn inequality, and derives the equivalence inf_n Λ^*_P(n) > 0 iff word-hyperbolicity by linking the spectral gap on dual graphs to the linear Dehn function (via uniform Cheeger constants on bounded-degree finite graphs). Area-minimizing diagrams exist by the well-ordering of positive integer areas and non-emptiness of the diagram set for each fixed boundary word. The free-completion and hole-free-ancestor construction is used solely to upgrade the profile to a QI-invariant, not to force the main characterization. No load-bearing step reduces to a self-citation, fitted parameter, or definitional tautology; the argument relies on external combinatorial facts and the new spectral objects.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Only the abstract is available, so the ledger is necessarily incomplete. The central claim rests on the existence of area-minimizing diagrams and standard facts about random-walk Laplacians on 2-complexes.

axioms (1)
  • domain assumption Every loop in a finitely presented group bounds a van Kampen diagram
    Invoked implicitly when defining Λ_P(n) via area-minimizing diagrams.
invented entities (1)
  • spectral Dehn function Λ_P(n) no independent evidence
    purpose: To serve as a spectral replacement for the classical Dehn function that detects word-hyperbolicity
    Newly defined as the infimum of the first Dirichlet eigenvalue over area-minimizing diagrams of boundary length ≤ n.

pith-pipeline@v0.9.0 · 5572 in / 1336 out tokens · 57490 ms · 2026-05-10T16:45:05.400445+00:00 · methodology

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Reference graph

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