Representation equivalent Bieberbach groups and strongly isospectral flat manifolds

Abstract

Let ${\Gamma }_1$ and ${\Gamma }_2$ be Bieberbach groups contained in the full isometry group $G$ of ${\mathbb{R}}^n$. We prove that if the compact flat manifolds ${\Gamma }_1\backslash {\mathbb{R}}^n$ and ${\Gamma }_2\backslash {\mathbb{R}}^n$ are strongly isospectral then the Bieberbach groups ${\Gamma }_1$ and ${\Gamma }_2$ are representation equivalent, that is, the right regular representations $L^2({\Gamma }_1\backslash G)$ and $L^2({\Gamma }_2\backslash G)$ are unitarily equivalent.

To appear in Canadian Mathematical Bulletin

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