Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds

Abstract

Let Γ1 and Γ2 be Bieberbach groups contained in the full isometry group G of Rn. We provethat if the compact flat manifolds Γ1\Rn and Γ2\Rn are strongly isospectral, then the Bieberbachgroups Γ1 and Γ2 are representation equivalent; that is, the right regular representations L2(Γ1\G) and L2(Γ2\G) are unitarily equivalent.