Isospectrality and 3-manifold Groups

In this note, we explain how a well-known construction of isospectral manifolds leads to an obstruction to a group being the fundamental group of a closed 3-dimensional manifold. The problem of determining, for a given group G, whether there is a closed 3-manifold M with π1(M) ∼= G is readily seen to be undecidable; let us write G ∈ G 3 if there is such a 3-manifold. A standard conjecture (related to Thurston’s geometrization program) states that the only possible finite groups in G are those which act freely and linearly on S, cf. [Tho86]. Algebraic ideas (which apply to the analogous problem in high-dimensional topology [DM85]) come close to proving this conjecture, although there are groups which are not ruled out by surgery theory but which do not act linearly. The new invariant we construct seems to be related to the classical surgery obstructions, but we do not know the precise relationship.