Profinite properties of graph manifolds

TL;DR

This paper investigates the profinite topology of fundamental groups of certain 3-manifolds, establishing efficiency, goodness, and conjugacy separability properties related to the JSJ decomposition.

Contribution

It proves the profinite topology is efficient for the JSJ decomposition, shows the fundamental group is good if all pieces are, and establishes conjugacy separability for graph manifolds.

Findings

Profinite topology is efficient with respect to JSJ decomposition.

Fundamental groups are good if all JSJ pieces are good.

Fundamental groups of graph manifolds are conjugacy separable.

Abstract

Let $M$ be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of $\pi_1(M)$ is efficient with respect to the JSJ decomposition of $M$. We go on to prove that $\pi_1(M)$ is good, in the sense of Serre, if all the pieces of the JSJ decomposition are. We also prove that if $M$ is a graph manifold then $\pi_1(M)$ is conjugacy separable.