Division Algebras and Non-Commensurable Isospectral Manifolds

2005/01/05 by Alexander Lubotzky, Lubotzky, Alexander, Beth Samuels +3
Mathematics · #11F72 #58J53 #Advanced Algebra and Geometry #Advanced Topics in Algebra #Algebraic structures and combinatorial models #FOS: Mathematics #Representation Theory (math.RT) #Spectral Theory (math.SP) #math.RT #math.SP #msc:11F72 #msc:58J53

paper · pdf · doi:10.48550/arxiv.math/0501064

22 pages

arxiv created 2005/01/05 · openalex publication_date 2005/01/05 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A. Reid showed that if Γ₁ and Γ₂ are arithmetic lattices in G = PGL₂(\mathbb R) or in PGL₂(\mathbb C) which give rise to isospectral manifolds, then Γ₁ and Γ₂ are commensurable (after conjugation). We show that for d ≥ 3 and \mathcal S = PGL_d(\mathbb R) / PGO_d(\mathbb R), or \mathcal S = PGL_d(\mathbb C) / 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.

Citations

• Isospectrality and commensurability of arithmetic hyperbolic 2- and 3-manifolds
• Can One Hear the Shape of a Drum?

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