Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds

LAURET, EMILIO A.

CANADIAN MATHEMATICAL BULLETIN-BULLETIN CANADIEN DE MATHEMATIQUES

CANADIAN MATHEMATICAL SOC

Año: 2013 vol. 57 p. 357 - 363

0008-4395

Let $\Gamma_1$ and $\Gamma_2$ be Bieberbach groups contained in the full isometry group $G$ of $\mathhb{R^n}$. 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 $L^2(\Gamma_1\backshalsh G)$ and $L^2(\Gamma_2\backshalsh G)$ are unitarily equivalent.