Lie Groups, Representations and Discrete Mathematics

In previous work we showed that arithmetic hyperbolic 2-manifolds that are isospectral are commensurable. In this talk we discuss the proof of the generalization to dimension 3. We had previously shown that if arithmetic hyperbolic 3-manifolds are complex iso-length spectral they are commensurable. What we will actually prove here is that arithmetic hyperbolic 3-manifolds that are iso-length spectral are commensurable. The proof has some surprising consequences for the galois theory of fields with one complex place.