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

Abstract

Let $\Gamma$ be a discrete finitely presented group. Pick any system $S$ of generators in $\Gamma$. In Cayley graph $\mathrm{Cay}(\Gamma)=\mathrm{Cay}(\Gamma, S)$ with edge set $E$, glue with oriented polygons all the group relations translated to all the points of $\Gamma$; denote the obtained simply connected complex by $\mathrm{Cay}^{(2)}(\Gamma)$. We study non-negative Hodge--Laplace operator $\Delta_1$ on edge functions which is defined via complex $\mathrm{Cay}^{(2)}(\Gamma)$; $\Delta_1$ acts on $$ \ell^2_{0,c}(E):= \mathrm{clos}_{\ell^2(E)} \left\{\mbox{finitely supported closed $1$-(co)chains in }\mathrm{Cay}^{}(\Gamma)\right\}. $$ We prove the following implication in the spirit of Kesten Theorem: if $\Delta_1|_{\ell_{0,c}^2(E)}$ has a spectral gap then $\Gamma$ either has exponential growth or is virtually $\mathbb Z$.