Unitary representations of fundamental groups and the spectrum of twisted Laplacians (Q1823479)

Let M be a closed Riemannian manifold and let \(\pi\) : \(X\to M\) be a regular cover with deck group G; as M is compact we may choose a finite set A generating G. Let \(\rho\) : \(G\to U(V)\) be a unitary representation on a separable Hilbert space V. Let \(\Delta_{\rho}\) be the associated Laplacian and let \(\lambda_ 0(\rho)\) be the greatest lower bound of the spectrum. Let \(\delta\) (\(\rho\),1) be the Kazhdan distance; \[ \delta (\rho,1)=\inf_{v\in V,| v| =1}\sup_{a\in A}| \rho (a)v- v|. \] Theorem 1. \(\exists c_ i(M,A)>0\) so \(\forall \rho\) \(c_ 1\delta (\rho,1)^ 2\leq \lambda_ 0(\rho)\leq c_ 2\delta (\rho,2)^ 2.\) This shows \(\lambda_ 0(\rho)=0\) \(\Leftrightarrow\) \(\delta (\rho,1)=0\). Let \(\rho\) be the regular representation. Then \(\delta (\rho,1)=0\) \(\Leftrightarrow\) G is amenable. This gives a proof of several results of Brooks: Corollary 2: (a) \(\lambda_ 0(X)=0\) \(\Leftrightarrow\) G is amenable. (b) If G satisfies Kazhdan property T, \(\exists c>0\) so every finite sub-covering \(M_ 1\to M\) of \(X\to M\), \(\lambda_ 1(M_ 1)>c\).