Profinite detection of JSJ decompositions of 3-manifolds

Gareth Wilkes

Published 2016-05-26Version 1

There has been much recent interest into those properties of a 3-manifold determined by the profinite completion of its fundamental group. In this paper we study graph manifolds and show that two graph manifolds with isomorphic profinite completions have the same JSJ decomposition in a certain strong sense. We also show that a pro-$p$ analogue of this theorem holds for certain classes of graph manifolds. Our results also distinguish graph manifolds among the class of all 3-manifolds and give information about the structure of totally hyperbolic manifolds.