Conjugacy problem in groups of oriented geometrizable 3-manifolds

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...

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“…Préaux [21] has shown that the conjugacy problem is solvable in the fundamental group of an orientable, geometrizable 3-manifold.…”

Section: Introductionmentioning

confidence: 99%

Profinite properties of graph manifolds

Wilton

Zalesskiĭ

2009

Geom Dedicata

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“…Préaux, extending Sela's work on knot groups [102], proved that the conjugacy problem is solvable, first for the fundamental groups of orientable 3-manifolds [90], and then also for the fundamental groups of non-orientable 3-manifolds [91]. (Note that, in contrast to many other group properties, solvability of the conjugacy problem does not automatically pass to finite extensions.)…”

Section: The Conjugacy Problemmentioning

confidence: 99%

Decision problems for 3-manifolds and their fundamental groups

Aschenbrenner

Friedl

Wilton

2015

Geometry &Amp; Topology Monographs

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We survey the status of some decision problems for 3-manifolds and their fundamental groups. This includes the classical decision problems for finitely presented groups (word problem, conjugacy problem, isomorphism problem), and also the homeomorphism problem for 3-manifolds and the membership problem for 3-manifold groups. 57M05 IntroductionThe classical group-theoretic decision problems were formulated by Max Dehn in his work on the topology of surfaces [23] about a century ago. He considered the following questions about finite presentations hA j Ri for a group :(1) The word problem, which asks for an algorithm to determine whether a word on the generators A represents the identity element of .(2) The conjugacy problem, which asks for an algorithm to determine whether two words on the generators A represent conjugate elements of .(3) The isomorphism problem, which asks for an algorithm to determine whether two given finite presentations represent isomorphic groups.Viewing as the fundamental group of a topological space (represented as a simplicial complex, say), these questions can be thought of as asking for algorithms to determine whether a given loop is null-homotopic, whether a given pair of loops is freely homotopic, and whether two aspherical spaces are homotopy-equivalent, respectively. We add some further questions that arise naturally: (4) The homeomorphism problem, which asks for an algorithm to determine whether two given triangulated manifolds are homeomorphic.(5) The membership problem, where the goal is to determine whether a given element of a group lies in a specified subgroup.Since the 1950s, it has been known that problems (1)- (5) , as a corollary of the unsolvability of (3).In contrast to this, in this paper we show that all these problems can now be solved for compact 3-manifolds and their fundamental groups, with the caveats that in (3) we restrict ourselves to closed, orientable 3-manifolds, and in (4) we restrict ourselves to orientable, irreducible 3-manifolds. Contrary to common perception (see [89, Section 7.1]) the homeomorphism problem for 3-manifolds is still open for reducible 3-manifolds. We discuss the status of the homeomorphism problem in detail in Section 4.5.The solutions to the first four problems, with the aforementioned caveats, have been known to the experts for a while, and as is to be expected, they all rely on the geometrization theorem. An algorithm for solving the membership problem was recently given in [28], and we will provide a summary of the main ideas in this paper. The solution to the membership problem requires not only the geometrization theorem but also the tameness theorem of Once a decision problem has been shown to be solvable, a natural next question concerns its complexity, and its implementability. The complexity of decision problems around 3-manifolds is a fascinating topic which is not touched upon in the present paper, except for a few references to the literature here and there. Practical implementations of algorithms for 3-manifolds are disc...

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“…As we mentioned in the introduction, Préaux, extending Sela's work on knot groups [Sel93], proved that the Conjugacy Problem is solvable for the fundamental groups of orientable [Pr06] and non-orientable [Pr12] 3-manifolds. (Note that, in contrast to many other group properties, solvability of the Conjugacy Problem does not automatically pass to finite extensions [CM77].…”

Section: The Word Problem and The Conjugacy Problem For 3-manifold Grmentioning

confidence: 99%