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Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds
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- Journal:
- Canadian Mathematical Bulletin / Volume 57 / Issue 2 / 14 June 2014
- Published online by Cambridge University Press:
- 20 November 2018, pp. 357-363
- Print publication:
- 14 June 2014
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- Article
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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 
${{L}^{2}}\left( {{\Gamma }_{1}}\backslash G \right)$ and 
${{L}^{2}}\left( {{\Gamma }_{2}}\backslash G \right)$ are unitarily equivalent.
