Spectral Dehn functions and a characterisation of word-hyperbolicity

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.