Virtually free pro- p groups whose torsion elements have finite centralizer

2009

Geom Dedicata

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...

Section: A Combination Theorem For Conjugacy Separable Groupsmentioning

confidence: 99%

Profinite properties of graph manifolds

Wilton

2009

Geom Dedicata

“…Apart from this not much is known about Aut(F n ), for example, it is not clear whether the virtual cohomological dimension of Aut(F n ) is finite. Certain progress however has been made in [4] and [5] towards understanding finite p-subgroups of Aut(F n ).…”

Section: Introductionmentioning

confidence: 99%

“…Let H be a cyclic group of order p r and let ρ : H → GL n (Z p ) be a representation. Then ρ lifts if and only if the Z p H-module M associated to ρ is a direct sum of indecomposable Z p H-modules which are isomorphic either to Z p C p k for some 0 ≤ k ≤ r or to J C p m (C p k ) for some 0 ≤ m < k ≤ r. The proofs of Theorem A and B are constructive in the sense that we give an explicit lifting φ : H → Aut(F n ) in terms of the semidirect product F n φ H. We use the result of [5] describing free-by-cyclic pro-p groups and a free decomposition of [4] for free-by-finite pro-p groups having finite centralizers of torsion elements. The representations in both theorems above are not necessarily faithful.…”

Section: Introductionmentioning

confidence: 99%

Free-by-finite pro-p groups and integral p-adic representations

Porto

2011

Arch. Math.

“…Define an equivalence relation on Σ by putting S 1 ∼ S 2 if S 1 and S 2 are contained in the same maximal finite subgroup of G 0 . This defines an equivalence relation since maximal finite subgroups have trivial intersection by [5,Lemma 9], and it is easy to see that ∼ is closed (indeed if K 1 , K 2 are maximal finite subgroups containing S 1 and S 2 respectively and such that K 1 ∩ K 2 = 1, then there is a finite quotient of G 0 where the images of K 1 and K 2 intersect trivially as well). Put T = Σ/ ∼.…”

Section: Hnn-embeddingmentioning

confidence: 99%

“…The description is a generalization of the main theorem of [5], where the result was obtained in the finitely generated case. Note however that finite generation is a rather restrictive condition in Galois theory and that the finite centralizer condition for torsion elements arises naturally in the study of maximal pro-p Galois groups.…”

Section: Introductionmentioning

confidence: 99%

Second countable virtually free pro-p groups whose torsion elements have finite centralizer

MacQuarrie

2016

Sel. Math. New Ser.

Self Cite

A second countable virtually free pro-p group all of whose torsion elements have finite centralizer is the free pro-p product of finite p-groups and a free pro-p factor. The proof explores a connection between p-adic representations of finite p-groups and virtually free pro-p groups. In order to utilize this connection, we first prove a version of a remarkable theorem of A. Weiss for infinitely generated profinite modules that allows us to detect freeness of profinite modules. The proof now proceeds using techniques developed in the combinatorial theory of profinite groups. Using an HNN-extension we embed our group into a semidirect product F ⋊ K of a free pro-p group F and a finite p-group K that preserves the conditions on centralizers and such that every torsion element is conjugate to an element of K. We then show that the ZpK-module F/[F, F ] is free using the detection theorem mentioned above. This allows us to deduce the result for F ⋊ K, and hence for our original group, using the pro-p version of the Kurosh subgroup theorem.