Spectral Dehn functions and a characterisation of word-hyperbolicity

Abstract: We introduce a \emph{spectral Dehn function} [ Λ{\mathcal{P}}(n):=\inf λ_1(Δ), ] where $λ_1(Δ)$ is the first Dirichlet eigenvalue of the random-walk Laplacian on a van Kampen diagram $Δ$, and the infimum runs over area-minimising diagrams with boundary length at most $n$. We prove a spectral-isoperimetric inequality relating $Λ</em>{\mathcal{P}}$ to the Dehn function, and show that its degree-free face-dual variant $Λ^{\ast_{\mathcal} P}$ characterises word-hyperbolicity: a finitely presented group is word-hyperbolic if and only if [ \inf_n Λ^\ast_{\mathcal{P}}(n)>0. ] Every disk diagram satisfies a diagramwise filling-length bound [ \mathrm{FL}_b(Δ)\cdot \operatorname{Area}(Δ) \ge c/λ_1(Δ); ] combined with a discrete Faber-Krahn inequality, this yields the sharp exponent $1/2$ in the quadratic case, attained by rectangular commutator grids over $\mathbb Z<sup>2$. 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.

• The paper introduces a spectral-analytic framework that refines classical Dehn functions to characterize word-hyperbolicity in finitely presented groups.
• It employs Laplacian spectral analysis on van Kampen diagrams, yielding sharp filling-length bounds through dual spectral/Dehn function comparisons.
• The method distinguishes group presentations with robust quasi-isometry invariants, addressing limitations of conventional isoperimetric criteria.

Spectral Dehn Functions and the Characterisation of Word-Hyperbolicity

Introduction and Context

The paper "Spectral Dehn functions and a characterisation of word-hyperbolicity" (2604.09014) revisits classical isoperimetric invariants in geometric group theory, introducing a spectral-analytic framework that marries Laplacian Dirichlet spectra of discrete planar complexes with the topological and combinatorial structure of van Kampen diagrams. This approach generates new criteria and invariants for word-hyperbolicity of finitely presented groups, culminating in several dual spectral/Dehn function characterisations and new quasi-isometry invariants.

Traditionally, the Dehn function $\delta_\mathcal{P}(n)$ captures the isoperimetric profile of a group presentation $\mathcal{P}$ and is central to the geometric theory of the word problem. However, the Dehn function alone can be insensitive to more subtle combinatorial or analytic phenomena, including quasi-isometry invariants and distinctions between presentations with linear (hyperbolic) behavior. The spectral Dehn framework provides both a refinement of this invariant and new, robust tools for analysing filling lengths and isoperimetric inequalities.

Definition and Main Properties of Spectral Dehn Functions

Let $\mathcal{P}$ be a finite presentation. For a van Kampen diagram $\Delta$ spanning a cyclically reduced null-homotopic word $w$, consider the first Dirichlet eigenvalue $\lambda_1(\Delta)$ of the random walk Laplacian on the $1$-skeleton of $\Delta$ (with absorbing boundary). The spectral Dehn function is defined as

$$\Lambda_\mathcal{P}(n) := \inf \{ \lambda_1(\Delta)\ :\ \Delta\text{ area-minimising},\ V^\circ(\Delta) \neq \varnothing,\ |\partial\Delta| \le n \}.$$

A degree-free, face-dual variant uses the first Dirichlet eigenvalue $\mu_1(\Delta)$ of the face-dual graph: $\mathcal{P}$0 These functions are inherently non-increasing in $\mathcal{P}$1. The face-dual formulation is particularly significant, as it provides a degree-free object and a canonical notion for comparing presentations, addressing limitations associated with primal graph vertex degrees.

A key result is the spectral-isoperimetric inequality: $\mathcal{P}$2 where $\mathcal{P}$3 is the minimal relator length.

Spectral Characterisation of Word-Hyperbolicity

The central claim is a new spectral characterisation of hyperbolic groups. A finitely presented group $\mathcal{P}$4 is word-hyperbolic if and only if its face-dual spectral Dehn function has a positive lower bound: $\mathcal{P}$5 This is a strong dichotomy: for non-hyperbolic groups, the best possible behavior is $\mathcal{P}$6—no intermediate decay is possible. Furthermore, no vertex degree bound is needed for $\mathcal{P}$7, in contrast to the primal profile.

The proof couples the spectral Cheeger inequality with encapsulation and surgery arguments on van Kampen diagrams, leveraging careful topological control (notably a hereditary quasi-minimality condition).

Filling-Length Rigidity and Spectral Collapse

The paper develops filling-length lower bounds in terms of spectral data. For a van Kampen diagram $\mathcal{P}$8 with area-minimal filling and $\mathcal{P}$9, the filling length $\mathcal{P}$0 at any basepoint $\mathcal{P}$1 obeys: $\mathcal{P}$2 When the Dehn function is polynomial, i.e. $\mathcal{P}$3 for area-minimising diagrams, one can sharpen this to

$\mathcal{P}$4

Through application of a discrete Faber–Krahn inequality, the exponent can be optimised to $\mathcal{P}$5 in the quadratic case ($\mathcal{P}$6), and this is shown to be sharp for grid diagrams over $\mathcal{P}$7.

This directly connects spectral collapse in fillings to geometric obstruction: vanishing Dirichlet eigenvalue enforces long, "thin" null-homotopies.

Euclidean Calibration: Sharpness of the $\mathcal{P}$8 Exponent

A precise computation for rectangular commutator grids over $\mathcal{P}$9 establishes that the filling length grows like $\Delta$0, with matching upper and lower bounds up to constants depending on the aspect ratio. This shows that, for diagrams with linear isoperimetric area, the $\Delta$1 spectral exponent for filling length is tight in both lower and upper bounds: $\Delta$2

Hereditarily Quasi-Minimality, Free Completion, and Quasi-Isometry Invariance

The minimal spectral Dehn function depends a priori on the presentation. To eliminate this dependence and obtain a quasi-isometry (QI) invariant, the paper introduces hereditarily quasi-minimal (HQM) disk maps, the free completion of complexes, and a hereditary hole-free-ancestor condition (hfmHQM). The free-completed hfmHQM profile—using disk maps over the universal cover with explicit free-cancellation bigons—gives a new profile whose positivity condition is a QI invariant of finitely presented groups and still fully characterises hyperbolicity.

For any two QI presentations $\Delta$3, the positivity of this spectral profile (i.e., $\Delta$4) is preserved under QI. The proof is based on bounded template replacement, pushforward constructions in bounded path-completions, and delicate diagram surgery, culminating in a quantitative interleaving result.

Finer Structure, Presentation Dependence, and Spectral Separation

The spectral Dehn functions are strictly finer than the standard Dehn class—they distinguish presentations even within the linear isoperimetric class. For example, free groups with cyclically reduced presentations admit diagrams with no $\Delta$5-cells (infinite spectral gap), whereas hyperbolic surface presentations with relators have constant spectral profile populated by the boundary relator disks ($\Delta$6).

Furthermore, for non-hyperbolic yet quadratically-filling groups (e.g., $\Delta$7), the spectral profile can be computed precisely and matches the scaling of grid Laplacians ($\Delta$8 for perimeter $\Delta$9).

Implications and Future Directions

The introduction of spectral Dehn functions opens a new interface between geometric group theory and discrete spectral analysis. The main contributions include:

• Effective duality between analytic and combinatorial invariants: The filling profile via face-dual eigenvalues captures both word-hyperbolicity and the geometry of fillings, defining a spectrum that is robust under presentation changes and quasi-isometry.
• Sharp filling-length lower bounds: The spectral methods yield optimal exponents, notably $w$0 for quadratic Dehn functions, and give sharp numeric bounds for explicit diagram families.
• Refined invariants and separation: Spectral Dehn functions can distinguish between presentations and group classes with the same classical Dehn function growth, suggesting new invariants that may detect subtle quasi-isometry class features.

There are several directions for future research:

• Extension to higher dimensions and different types of fillings: Can analogues be constructed for higher-order filling invariants or non-orientable surfaces?
• Structural comparison and universality: Investigating whether the HQM and hfmHQM profiles are coarsely equivalent to the minimal profile, and how these quantities behave under further group extensions or even for groups with Dehn function $w$1 with $w$2.
• Spectral separation in non-linear Dehn classes: Can one construct examples (within the quadratic or cubic classes) showing that spectral profiles can separate groups within the same Dehn class but with different filling geometries?
• Algorithmic and computational aspects: Are there efficient algorithms for approximating spectral Dehn functions given a presentation? Can these invariants be leveraged in practical group-theoretic computations or in the study of random groups?

Conclusion

"Spectral Dehn functions and a characterisation of word-hyperbolicity" (2604.09014) formulates and analyses spectral invariants of van Kampen diagrams, delivering tight algebraic and geometric characterisations of hyperbolicity. It establishes a quasi-isometry invariant spectral criterion and achieves sharp spectral-filling length lower bounds, distinguishing presentations even within the same Dehn growth class. The methods provide new tools for the analytic investigation of isoperimetric and combinatorial group invariants and set a foundation for further exploration into the spectral geometry of discrete filling problems.