Duke Mathematical Journal

Division algebras and noncommensurable isospectral manifolds

Alexander Lubotzky, Beth Samuels, and Uzi Vishne

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Abstract

A. W. Reid [R, Theorem 2.1] showed that if Γ1 and Γ2 are arithmetic lattices in G=PGL2(R) or in PGL2(C) which give rise to isospectral manifolds, then Γ1 and Γ2 are commensurable (after conjugation). We show that for d3 and S=PGLd(R)/POd(R) or for S=PGLd(C)/PUd(C), the situation is quite different; there are arbitrarily large finite families of isospectral noncommensurable compact manifolds covered by S. The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants

Article information

Source
Duke Math. J., Volume 135, Number 2 (2006), 361-379.

Dates
First available in Project Euclid: 17 October 2006

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1161093268

Digital Object Identifier
doi:10.1215/S0012-7094-06-13525-1

Mathematical Reviews number (MathSciNet)
MR2267287

Zentralblatt MATH identifier
1123.58020

Subjects
Primary: 58J35: Heat and other parabolic equation methods
Secondary: 11F70: Representation-theoretic methods; automorphic representations over local and global fields

Citation

Lubotzky, Alexander; Samuels, Beth; Vishne, Uzi. Division algebras and noncommensurable isospectral manifolds. Duke Math. J. 135 (2006), no. 2, 361--379. doi:10.1215/S0012-7094-06-13525-1. https://projecteuclid.org/euclid.dmj/1161093268


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