Twisted isospectrality, homological wideness and isometry

Abstract

Given a manifold (or, more generally, a developable orbifold) $M_0$ and two closed Riemannian manifolds $M_1$ and $M_2$ with a finite covering map to $M_0$, we give a spectral characterisation of when they are equivalent Riemannian covers (in particular, isometric), assuming a representation-theoretic condition of "homological wideness": if $M$ is a common finite cover of $M_1$ and $M_2$ and $G$ is the covering group of $M$ over $M_0$, the condition involves the action of $G$ on the first homology group of $M$ (it holds, for example, when there exists a rational homology class on $M$ whose orbit under $G$ consists of $|G|$ linearly independent homology classes). We prove that, under this condition, Riemannian covering equivalence is the same as isospectrality of finitely many twisted Laplacians on the manifolds, acting on sections of flat bundles corresponding to specific representations of the fundamental groups of the manifolds involved. Using the same methods, we provide spectral criteria for weak conjugacy and strong isospectrality. In the negative curvature case, we formulate an analogue of our result for the length spectrum. The proofs are inspired by number-theoretical analogues. We study examples where the representation theoretic condition does and does not hold. For example, when $M_1$ and $M_2$ are commensurable non-arithmetic closed Riemann surfaces of negative Euler characteristic, there is always such an $M_0$, and the condition of homological wideness always holds.