Division Algebras and Non-Commensurable Isospectral Manifolds

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dc.creatorLubotzky, Alexander
dc.creatorSamuels, Beth
dc.creatorVishne, Uzi
dc.date2005-01-05
dc.date.accessioned2026-07-07T05:15:50Z
dc.date.available2026-07-07T05:15:50Z
dc.descriptionA. Reid showed that if $Γ_1$ and $Γ_2$ are arithmetic lattices in $G = \operatorname{PGL}_2(\mathbb R)$ or in $\operatorname{PGL}_2(\mathbb C)$ which give rise to isospectral manifolds, then $Γ_1$ and $Γ_2$ are commensurable (after conjugation). We show that for $d \geq 3$ and ${\mathcal S} = \operatorname{PGL}_d(\mathbb R) / \operatorname{PGO}_d(\mathbb R)$, or ${\mathcal S} = \operatorname{PGL}_d(\mathbb C) / \operatorname{PU}_d(\mathbb C)$, the situation is quite different: there are arbitrarily large finite families of isospectral non-commensurable compact manifolds covered by $\mathcal S$. The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants.
dc.description22 pages
dc.identifierhttps://arxiv.org/abs/math/0501064
dc.identifierhttp://arxiv.org/abs/math/0501064
dc.identifier.urihttp://salesiana.dossiersoluciones.com/handle/123456789/73769
dc.subjectSpectral Theory
dc.subjectRepresentation Theory
dc.subject58J53, 11F72
dc.titleDivision Algebras and Non-Commensurable Isospectral Manifolds
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