Profinite properties of graph manifolds

LetM be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of π1(M) is efficient with respect to the JSJ decomposition of M . We go on to prove that π1(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 π1(M) is conjugacy separable. A group G is conjugacy separable if every conjugacy class is closed in the profinite topology on G. This can be thought of as a strengthening of residual finiteness (which is equivalent to the trivial subgroup’s being closed). Hempel [9] proved that the fundamental group of any geometrizable 3-manifold is residually finite. In this paper, we investigate which 3-manifolds have conjugacy separable fundamental group. We also study Serre’s notion of goodness, another property related to the profinite topology. Let M be a compact, connected 3-manifold. Let D be the closed 3manifold obtained by doubling M along its boundary. The inclusion M →֒ D has a natural left inverse. At the level of fundamental groups it follows that π1(M) injects into π1(D) and that two elements are conjugate in π1(M) if and only if they are conjugate in π1(D). Hence, if π1(D) is conjugacy separable then so is π1(M). Therefore, we can assume that M is closed. Because conjugacy separability is preserved by taking free products [27], we may take M to be irreducible. As a technical assumption, we shall also ∗Partially supported by CNPq. A conjugacy separable group has solvable conjugacy problem. Préaux [20] has shown that the conjugacy problem is solvable in the fundamental group of an orientable, geometrizable 3-manifold.