Isospectrality and commensurability of arithmetic hyperbolic 2- and 3- manifolds (Q1187733)

The known methods of constructing isospectral but nonisometric hyperbolic 2- and 3-manifolds produce manifolds which are commensurable. It is an open question as to whether isospectrality implies commensurability. The main result of the paper under review gives an interesting partial answer to this question: Any two isospectral nonisometric arithmetic hyperbolic 2- or 3-manifolds are commensurable. Taking up a theme developed by Spatzier, the author also proves that isospectral but nonisometric hyperbolic manifolds are to be found in great abundance in dimension 3.