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# Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds

Published:2013-05-26
Printed: Jun 2014
• Emilio A. Lauret,
Facultad de Matemática Astronomía y Física (FaMAF), Universidad Nacional de Córdoba, Medina Allende s/n, Ciudad Universitaria, X5000HUA, Córdoba, Argentina.
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## 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.
 Keywords: representation equivalent, strongly isospectrality, compact flat manifolds
 MSC Classifications: 58J53 - Isospectrality 22D10 - Unitary representations of locally compact groups

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