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Symmetry, Integrability and Geometry: Methods and Applications, 2011, Volume 7, 080, 8 pp.
DOI: https://doi.org/10.3842/SIGMA.2011.080
(Mi sigma638)
 

This article is cited in 1 scientific paper (total in 1 paper)

The 2-Transitive Transplantable Isospectral Drums

Jeroen Schillewaerta, Koen Thasb

a Department of Mathematics, Free University of Brussels (ULB), CP 216, Boulevard du Triomphe, B-1050 Brussels, Belgium
b Department of Mathematics, Ghent University, Krijgslaan 281, S25, B-9000 Ghent, Belgium
Full-text PDF (329 kB) Citations (1)
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Abstract: For Riemannian manifolds there are several examples which are isospectral but not isometric, see e.g. J. Milnor [Proc. Nat. Acad. Sci. USA 51 (1964), 542]; in the present paper, we investigate pairs of domains in $\mathbb R^2$ which are isospectral but not congruent. All known such counter examples to M. Kac's famous question can be constructed by a certain tiling method (“transplantability”) using special linear operator groups which act $2$-transitively on certain associated modules. In this paper we prove that if any operator group acts $2$-transitively on the associated module, no new counter examples can occur. In fact, the main result is a corollary of a result on Schreier coset graphs of $2$-transitive groups.
Keywords: isospectrality; drums; Riemannian manifold; doubly transitive group; linear group.
Received: December 14, 2010; in final form August 8, 2011; Published online August 18, 2011
Bibliographic databases:
Document Type: Article
Language: English
Citation: Jeroen Schillewaert, Koen Thas, “The 2-Transitive Transplantable Isospectral Drums”, SIGMA, 7 (2011), 080, 8 pp.
Citation in format AMSBIB
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  • This publication is cited in the following 1 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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