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Representation Equivalent Bieberbach Groups and Strongly Isospectral Flat Manifolds

  • Emilio A. Lauret (a1)
Abstract

Let Γ1 and Γ2 be Bieberbach groups contained in the full isometry group G of ℝ n . We prove that if the compact flat manifolds Γ1\ℝn and Γ2\ℝn are strongly isospectral, then the Bieberbach groups Γ1 and Γ2 are representation equivalent; that is, the right regular representations L 21\G) and L 22\G) are unitarily equivalent.

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[Go09] Gordon, C., Sunada's isospectrality technique: two decades later. In: Spectral analysis in geometry and number theory, Contemp. Math., 484, American Mathematical Society, Providence, RI, 2009), pp. 4558.
[LMR12] Lauret, E., Miatello, R., and Rossetti, J. P., Representation equivalence and p-spectrum of constant curvature space forms. J. Geom. Anal., to appear; arxiv:1209.4916[math.SP].
[Pe95] Pesce, H., Variétés hyperboliques et elliptiques fortement isospectrales. J. Funct. Anal. 133 (1995), no. 2, 363391.http://dx.doi.org/10.1006/jfan.1995.1150
[Pe96] Pesce, H., Représentations relativement équivalentes et variétés riemanniennes isospectrales. Comment. Math. Helv. 71 (1996), no. 2, 243268.http://dx.doi.org/10.1007/BF02566419
[Wa73] Wallach, N. R., Harmonic analysis on homogeneous spaces. Pure and AppliedMathematics, 19, Marcel Dekker, Inc., New York, 1973.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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