Representation equivalent Bieberbach groups and strongly isospectral flat manifolds

Abstract

Let $Γ_1$ and $Γ_2$ be Bieberbach groups contained in the full isometry group $G$ of $\mathbb{R}^n$. We prove that if the compact flat manifolds $Γ_1\backslash\mathbb{R}^n$ and $Γ_2\backslash\mathbb{R}^n$ are strongly isospectral then the Bieberbach groups $Γ_1$ and $Γ_2$ are representation equivalent, that is, the right regular representations $L^2(Γ_1\backslash G)$ and $L^2(Γ_2\backslash G)$ are unitarily equivalent.