Abstract: There has been recent interest in those properties of 3-manifolds which can be characterised by the behaviour of their finite quotients. In particular one may study the circumstances under which two non-homeomorphic 3-manifolds may have fundamental groups with the same finite quotients. I will describe the classification of those pairs of graph manifolds with this property, and illustrate by means of examples the way in which such manifolds may arise. I will also discuss the implications of this result for the mapping class group.
Profinite rigidity of graph manifolds