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DIVISION ALGEBRAS AND NON-COMMENSURABLE ISOSPECTRAL MANIFOLDS

A. Reid [R] showed that if Γ1 and Γ2 are arithmetic lattices in G = PGL2(R) or in PGL2(C) which give rise to isospectral manifolds, then Γ1 and Γ2 are commensurable (after conjugation). We show that for d ≥ 3 and S = PGLd(R)/POd(R), or S = PGLd(C)/PUd(C), the situation is quite different: there are arbitrarily large finite families of isospectral non-commensurable compact manifolds covered by S. The constructions are based on the arithmetic groups obtained from division algebras with the same ramification points but different invariants.