Division algebras and noncommensurable isospectral manifolds

Abstract

A. W. Reid [R, Theorem 2.1] 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 $\mathcal{S}={\mathrm{PGL}}_d(\mathbb{R})/{\mathrm{PO}}_d(\mathbb{R})$ or for $\mathcal{S}={\mathrm{PGL}}_d(\mathbb{C})/{\mathrm{PU}}_d(\mathbb{C})$, the situation is quite different; there are arbitrarily large finite families of isospectral noncommensurable 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