Profinite rigidity of graph manifolds and JSJ decompositions of 3-manifolds

Abstract

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 give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish graph manifolds among the class of all 3-manifolds and give information about the structure of totally hyperbolic manifolds, and give control over the pro-$p$ completion of certain graph manifold groups.

arXiv comment48 pages. This is an improved and extended version of a previous paper `Profinite detection of JSJ decompositions of 3-manifolds'. The new version includes, in addition to the previous results, the theorem characterising when two graph manifold groups have isomorphic profinite completions