Spectral Invariance and Gevrey Regularity for Groups with strongly subexponential growth

Abstract

We study spectral invariance and Gevrey regularity for convolution operators with kernels in suitable weighted function spaces on locally compact groups equipped with a locally bounded length function $\ell$. The main analytic scale is given by the subexponential weights. For groups whose volume growth is bounded above by $e^{R^\gamma}$ for some $0<\gamma<1$, we establish spectral comparison result for compactly supported functions. For compactly supported Hermitian functions, we prove spectral radius invariance across the symmetric $q$-pseudofunction $*$-algebra, the weighted and unweighted group algebras, and the full and reduced group $C^*$-algebras. For unimodular groups satisfying strong subexponential growth of exponent at most $\beta$, we construct a Gevrey-Beurling operator algebra inside the unitized $q$-pseudofunction algebra. We prove that this algebra is inverse-closed and that its inclusion induces an isomorphism in topological $K$-theory. The inverse-closedness theorem may be viewed as a quantitative Gevrey-type noncommutative Wiener lemma. As an application, we show that whenever a convolution operators with kernels in the corresponding weighted Gevrey-Beurling space is invertible in the unitized $q$-pseudofunction algebra, then its inverse belongs to the same Gevrey-Beurling operator algebra and satisfies explicit Gevrey seminorm estimates. We also develop a relative theory for pairs of finitely generated groups using Schreier graph lengths and quasi-regular representations. This provides a subexponential analogue of rapid decay for group pairs, when subgroup is normal, it reduces to the usual theory on the quotient. The framework can apply to intermediate-growth examples, including the Grigorchuk group, and is stable under products with polynomial growth groups and under compact extensions.