Conjugacy Separability of Amalgamated Free Products of Groups

Ribes, Luis; Zalesskiĭ, Pavel

doi:10.1006/jabr.1996.0035

Cited by 36 publications

(43 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Let p be the epimorphism from π 1 (M) to π 1 (O). Since π 1 (O) is virtually free, the intersection of p(C 1 ) and p(C 2 ) is trivial since the intersection of p(C 1 ) ∩ p(C 2 ) is (see Lemma 3.6 in [24]), so the intersection of C 1 and C 2 is Z.…”

Section: Proofmentioning

confidence: 99%

Profinite properties of graph manifolds

Wilton

Zalesskiĭ

2009

Geom Dedicata

View full text Add to dashboard Cite

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...

show abstract

Section: Proofmentioning

confidence: 99%

Profinite properties of graph manifolds

Wilton

Zalesskiĭ

2009

Geom Dedicata

View full text Add to dashboard Cite

show abstract

“…Then, by [7,Lemma 2.8], the element b also stabilizes a vertex of S(G). This means that a and b are conjugate in G with some elements of G1 or G2.…”

Section: Case 1 the Element A Stabilizes A Vertex Of The Graph S(g)mentioning

confidence: 93%

“…Let G = Gx *H G2 be the amalgamated free product of groups G1 and G2 by the subgroup H. Dyer [6] showed that if both Ga and G2 are free or both are finitely generated nilpotent groups and if H is cyclic, then G = Ga *H G2 is finitely approximable with respect to conjugacy. Recently Ribes and Zalesskii [7] generalized this result to the case in which Ga and G~ are almost free or almost finitely generated nilpotent groups (not necessarily of the same type) and H is cyclic.…”

mentioning

confidence: 90%

Closed orbits and finite approximability with respect to conjugacy of free amalgamated products

Zalesskiĭ

Tavgen

1995

Math Notes

View full text Add to dashboard Cite

Let G be a group. We say that G is finitely approzimable with respect to co•jugacy if any two elements conjugate in each finite quotient group of the group G are conjugate in G. The topology on G whose basis of neighborhoods of the unit element is the set of all normal subgroups of finite index is called the profinite topology. Then the group G is finitely approximable with respect to conjugacy if the conjugacy class of each element of G is closed in the profinite topology.The importance of this notion is shown by the result of Mal'tsev [1], which states that if a finitely determined group G is finitely approximable with respect to conjugacy, then G admits an affirmative solution of the conjugacy problem, i.e., there exists an algorithm that determines whether two given elements are conjugate.Remeslennikov [2] and Formanek [3] proved that almost polycyclic groups are finitely approximable with respect to conjugacy. Recall that a group is said to "almost" have a given property if this group contains a subgroup of finite index with this property. We note that it is still unknown whether a group that is almost finitely approximable with respect to conjugacy must be itself finitely approximable with respect to conjugacy. Further, Stebe [4] and Remeslennikov [5] showed that the free product of groups that are finitely approximable with respect to conjugacy is itself finitely approximable with respect to conjugacy.The extension of these results to amalgamated free products is a significantly more complicated problem. One of the reasons is in the fact that if a group G is finitely approximable with respect to conjugacy, then it is finitely approximable. However, not e'r amalgamated free product of finitely approximable groups is finitely approximable. Therefore, the problem of finite approximability with respect to conjugacy makes sense only for finitely approximable amalgamated free products.

show abstract

“…Observe that the profinite completion N of Π has exactly the same presentation and so N F 1 * F 2 is a profinite amalgamated free product of two finitely generated free profinite groups with a procyclic amalgamated subgroup. In fact, this profinite amalgamated free product is proper (see [11,Proposition 3.2]) in the sense that F 1 , Z and F 2 are naturally embedded in N .…”

Section: There Is An Open Normal Subgroupmentioning

confidence: 99%

Normal Subgroups of Profinite Groups of Finite Cohomological Dimension

Engler

Haran

Kochloukova

et al. 2004

J. London Math. Soc.

View full text Add to dashboard Cite

We study a profinite group G of finite cohomological dimension with (topologically) finitely generated closed normal subgroup N . If G is pro-p and N is either free as a pro-p group or a Poincaré group of dimension 2 or analytic pro-p, we show that G/N has virtually finite cohomological dimension cd(G) − cd(N ). Some other cases when G/N has virtually finite cohomological dimension are considered too.If G is profinite, the case of N projective or the profinite completion of the fundamental group of a compact surface is considered.

show abstract

Conjugacy Separability of Amalgamated Free Products of Groups

Cited by 36 publications

References 21 publications

Profinite properties of graph manifolds

Profinite properties of graph manifolds

Closed orbits and finite approximability with respect to conjugacy of free amalgamated products

Normal Subgroups of Profinite Groups of Finite Cohomological Dimension

Contact Info

Product

Resources

About