Profinite properties of graph manifolds

Abstract

Let M be a closed, orientable, irreducible, geometrizable 3-manifold. We prove that the profinite topology on the fundamental group of π 1 (M ) is efficient with respect to the JSJ decomposition of M . We go on to prove that π 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 π 1 (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.1 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 3-manifold 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 π 1 (M) injects into π 1 (D) and that two elements are conjugate in π 1 (M) if and only if they are conjugate in π 1 (D). Hence, if π 1 (D) is conjugacy separable then so is π 1 (M). Therefore, we can assume that M is closed.Because conjugacy separability is preserved by taking free products [28], we may take M to be irreducible. As a technical assumption, we shall also * Partially supported by CNPq.1 A conjugacy separable group has solvable conjugacy problem. Préaux [21] has shown that the conjugacy problem is solvable in the fundamental group of an orientable, geometrizable 3-manifold. assume that M is orientable.2 Under these hypotheses, M has a canonical JSJ decomposition, the pieces of which are either Seifert-fibred or, according to the Geometrization Conjecture, admit finite-volume hyperbolic structures. By the Seifert-van Kampen Theorem, the JSJ decomposition of M induces a graph-of-groups decomposition of the fundamental group.Our first theorem asserts that this graph of groups is, from a profinite point of view, well behaved. If a residually finite group G is the fundamental group of a graph of groups (G, Γ), the profinite topology on G is called efficient if the vertex and edge groups of G are closed and if the profinite topology on G induces the full profinite topologies on the vertex and edge groups of G.Theorem A Let M be a closed, orientable, irreducible, geometrizable 3-manifold, and let (G, Γ) be the graph-of-groups decomposition of π 1 (M) induced by the JSJ decomposition of M. Then the profinite topology on π 1 (M) is efficient.Theorem A provides the foundation for our main theorems, which relate the profinite completion of π 1 (M) to the profinite completions of the pieces of the JSJ decomposition.Our next theorem is a digression regarding goodness, a property introduced by Serre (see I.2.6 Exercise 2 in [27]). A group G is good if the natural map from G to its profinite completion i...

$3$-Manifold Groups are Virtually Residually $p$

“…This complements a result by Wilton and Zalesskii [WiltZa10] who showed that every graph manifold group G is good in the sense of Serre [Se97, Section 2.6, exercises], that is, with G denoting the profinite completion of G, for every finite G-module M and n ≥ 1, the natural morphism G → G induces an isomorphism H Lemma 5.11. Suppose G is a p-efficient graph of finitely generated groups with underlying graph Y and fundamental group G = π 1 (G).…”

“…It seems plausible that this result might play the same role in studying the (virtual) conjugacy p-separability of graph manifold groups as finite efficiency does in the proof of their conjugacy separability, which was also established in [WiltZa10]. We do not pursue this issue further in the present paper; however, we note that Proposition 3 also implies that virtually, the fundamental group of a closed graph manifold group and its pro-p completion have the same mod p cohomology in a certain sense, explained in Section 5.2.…”

“…Wilton and Zalesskii [WiltZa10] showed that all closed prime orientable 3-manifolds are P-efficient, for P the property of being finite. We show: Proposition 3.…”

Hereditary conjugacy separability of free products with amalgamation

“…Note that an amalgamated free product is a special case of this situation, namely corresponding to the case when Γ consists of one edge and two vertices. The next proposition from [22] collects (from [17]) some useful facts about group actions on profinite trees and will be used frequently in this paper. We note however that it does not need full strength of the hypothesis there, but only the embedding S(G) into S(  G) that in turn follows from closedness of vertex and edge groups in the profinite topology.…”

Grothendieck’s problem for 3-manifold groups

“…(4) Lemma 5.3 is in the spirit of a result of Wilton and Zalesskii [49], where it is proved that fundamental groups of the vertex manifolds in the JSJ decomposition are separable.…”