Division Algebras and Non-Commensurable Isospectral Manifolds

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Lubotzky, Alexander; Samuels, Beth; Vishne, Uzi;
(2005)
  • Subject: Mathematics - Spectral Theory | Mathematics - Representation Theory | 58J53, 11F72
    arxiv: Mathematics::Representation Theory | Mathematics::Algebraic Geometry | Mathematics::Geometric Topology | Mathematics::Group Theory | Mathematics::Probability

A. Reid showed that if $\Gamma_1$ and $\Gamma_2$ are arithmetic lattices in $G = \operatorname{PGL}_2(\mathbb R)$ or in $\operatorname{PGL}_2(\mathbb C)$ which give rise to isospectral manifolds, then $\Gamma_1$ and $\Gamma_2$ are commensurable (after conjugation). We s... View more
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    By Theorem 3, L2(G′1(k)\G0) ∼= L2(G′2(k)\G0) as representation spaces, so the result follows.

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