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Lubotzky, Alexander; Samuels, Beth; Vishne, Uzi

Division algebras and noncommensurable isospectral manifolds. (English) Zbl 1123.58020

Duke Math. J. 135, No. 2, 361-379 (2006).
Author’s abstract: A. W. Reid showed in [Duke Math. J. 65, No. 2, 215–228 (1992; Zbl 0776.58040)] that if \(\Gamma_1\) and \(\Gamma_2\) are arithmetic lattices in \(G=\text{PGL}_2(\mathbb R)\) or in \(G=\text{PGL}_2(\mathbb C)\) which give rise to isospectral manifolds, then \(\Gamma_1\) and \(\Gamma_2\) are commensurable (after conjugation). We show that for \(d\geq 3\) and \( \mathcal {S}=\text{PGL}_d(\mathbb R)/\text{PO}_d(\mathbb R)\) or \( \mathcal {S}=\text{PGL}_d(\mathbb C)/\text{PO}_d(\mathbb C)\), the situation is quite different; there are arbitrarily large finite families of isospectral noncommensurable compact manifolds covered by \(\mathcal S\). The constructions are based on the arithmetic groups obtained from divisions algebras with the same ramification points but different invariants.
Reviewer: Witold Mozgawa (Lublin)

Cited in 10 Documents

MSC:

58J53 Isospectrality
11F70 Representation-theoretic methods; automorphic representations over local and global fields
17A35 Nonassociative division algebras

Keywords:

isospectral; virtually isomorphic; arithmetic lattice; noncommensurability; division algebra

Citations:

Zbl 0776.58040
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Full Text: DOI arXiv

References:

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