Coexact 1-Laplacian spectral gap and exponential growth of a group
Mikhail Dubashinskiy
TL;DR
The paper links the spectral properties of the coexact 1-Laplacian $\Delta_1$ on the simply connected Cayley 2-complex of a finitely presented group to the group’s growth. By embedding long controlled loops into Cayley graphs and employing a resolvent-approximation strategy, it establishes a Kesten-like dichotomy: if $\Delta_1$ has a spectral gap on $\ell^2_{0,c}(E)$ then $\Gamma$ exhibits exponential growth or is virtually $\mathbb{Z}$. The approach leverages a functional-analytic framework with loop- and cochain-based decompositions, proving the invariance of the spectral-gap property under changing generating sets and illustrating the theory with concrete examples: $\mathbb{Z}^m$ (no gap for $m\ge2$) and genus-$\mathfrak g$ surface groups (gap via a Riesz-basis of cycles). The results connect spectral gaps to algebraic growth, offering a pathway to understand isoperimetric-type phenomena in discrete groups through 2-dimensional complexes.
Abstract
Let $Γ$ be a discrete finitely presented group. Pick any system $S$ of generators in $Γ$. In Cayley graph $\mathrm{Cay}(Γ)=\mathrm{Cay}(Γ, S)$ 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 $\mathrm{Cay}^{(2)}(Γ)$. We study non-negative Hodge--Laplace operator $Δ_1$ on edge functions which is defined via complex $\mathrm{Cay}^{(2)}(Γ)$; $Δ_1$ acts on $$ \ell^2_{0,c}(E):= \mathrm{clos}_{\ell^2(E)} \left\{\mbox{finitely supported closed $1$-(co)chains in }\mathrm{Cay}^{}(Γ)\right\}. $$ We prove the following implication in the spirit of Kesten Theorem: if $Δ_1|_{\ell_{0,c}^2(E)}$ has a spectral gap then $Γ$ either has exponential growth or is virtually $\mathbb Z$.