Division Algebras and Non-Commensurable Isospectral Manifolds

Abstract

A. Reid showed that if ${\Gamma }_1$ and ${\Gamma }_2$ are arithmetic lattices in $G={\mathrm{PGL}}_2(\mathbb{R})$ or in ${\mathrm{PGL}}_2(\mathbb{C})$ which give rise to isospectral manifolds, then ${\Gamma }_1$ and ${\Gamma }_2$ are commensurable (after conjugation). We show that for $d\ge 3$ and $\mathcal{S}={\mathrm{PGL}}_d(\mathbb{R})/{\mathrm{PGO}}_d(\mathbb{R})$, or $\mathcal{S}={\mathrm{PGL}}_d(\mathbb{C})/{\mathrm{PU}}_d(\mathbb{C})$, 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.

22 pages