Coverings preserving the bottom of the spectrum (Q6039735)

A right action of a countable group \(\Gamma\) on a countable set \(X\) is called amenable if there is a \(\Gamma\)-invariant mean on \(L_\infty(X)\). Let \(M_1\) and \(M_2\) be two Riemannian manifolds, and \(p\) be a \(M_1\)-valued Riemannian covering on \(M_2\), we say that \(p\) is amenable if the right action of \(\pi_1(M_1)\) (the fundamental group of \(M_1\) with base point \(x\) in the interior of \(M_1\)) on \(p^{-1}(x)\) is amenable. We denote by \(\lambda_0\) the bottom of spectrum. The author states the following theorem. Let \(p:M_2\to M_1\) be a Riemannian covering, \(S_1\) be a Schrödinger operator on \(M_1\), with \(\lambda_0(S_1)\notin \sigma_{\mathrm{ess}}(S_1)\), and \(S_2\) be its lift on \(M_2\). Then \(\lambda_0(S_2)=\lambda_0(S_1)\) if and only if \(p\) is amenable.