Homological characterisation of Lambda-ranks

Twisted isospectrality, homological wideness and isometry

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 orb…