Coexact 1-Laplacian spectral gap and exponential growth of a group

2025/11/04 by Dubashinskiy, Mikhail
#53C23 #FOS: Mathematics #Group Theory (math.GR) #Metric Geometry (math.MG) #Primary: 58J50 #Secondary: 20F65 #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2511.02138

Abstract

Let Γ be a discrete finitely presented group. Pick any system S of generators in Γ. In Cayley graph Cay(Γ)=Cay(Γ, S) with edge set E, glue with oriented polygons all the group relations translated to all the points of Γ; denote the obtained simply connected complex by Cay⁽²⁾(Γ). We study non-negative Hodge--Laplace operator Δ₁ on edge functions which is defined via complex Cay⁽²⁾(Γ); Δ₁ acts on ℓ²_0,c(E):= clos_ℓ²(E) \finitely supported closed 1-(co)chains in Cay(Γ)\. We prove the following implication in the spirit of Kesten Theorem: if Δ₁|_ℓ_0,c²(E) has a spectral gap then Γ either has exponential growth or is virtually \mathbb Z.

Citations

• Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms
• A Cheeger-like inequality for coexact 1-forms
• Stable Isoperimetric Ratios and the Hodge Laplacian of Hyperbolic Manifolds
• Discretization of Riemannian manifolds applied to the Hodge Laplacian

Related

• On the spectral gap of some Cayley graphs on the Weyl group W(B_n)
• On spectra and spectral measures of Schreier and Cayley graphs
• Eigenvalue multiplicity and volume growth
• On a cheeger type inequality in Cayley graphs of finite groups
• Laplace and Schrödinger operators without eigenvalues on homogeneous amenable graphs