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

TL;DR

This paper establishes criteria to determine when two orientable graph manifold groups have isomorphic profinite completions, aiding in distinguishing graph manifolds and understanding their structure within 3-manifolds.

Contribution

It provides new, computable criteria for profinite rigidity of graph manifolds and insights into their JSJ decompositions and hyperbolic structures.

Findings

Criteria for profinite isomorphism of graph manifold groups

Distinguishing graph manifolds among all 3-manifolds

Insights into the pro-p completions of certain graph manifold groups

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.