Residual finiteness, QCERF and fillings of hyperbolic groups

Agol, Ian; Groves, Daniel; Manning, Jason Fox

doi:10.2140/gt.2009.13.1043

Cited by 41 publications

(82 citation statements)

References 18 publications

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“…Fully quasiconvex subgroups appear in the work of F. Dahmani [14] where it is shown that, under some hypothesis, the combination of relatively hyperbolic groups along fully quasiconvex subgroups is a relatively hyperbolic group.This class of subgroups also appeared in the the work by J. Manning and the author [22] to prove a consequence of the hypothetical absence of non-residually finite hyperbolic groups that was conjectured by I. Agol, D. Groves and J. Manning [1].…”

Section: Sample Applicationsmentioning

confidence: 71%

Combination of quasiconvex subgroups of relatively hyperbolic groups

Pedroza

2009

Groups Geom. Dyn.

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Section: Sample Applicationsmentioning

confidence: 71%

Combination of quasiconvex subgroups of relatively hyperbolic groups

Pedroza

2009

Groups Geom. Dyn.

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“…We will also need to make use of a relatively hyperbolic extension of the results of Agol, Groves and Manning from [AGM08].…”

Section: Malnormality In the Relative Casementioning

confidence: 99%

Distinguishing geometries using finite quotients

Wilton

Zalesskiĭ

2017

Geom. Topol.

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We prove that the profinite completion of the fundamental group of a compact 3-manifold M satisfies a Tits alternative: if a closed subgroup H does not contain a free pro-p subgroup for any p, then H is virtually soluble, and furthermore of a very particular form. In particular, the profinite completion of the fundamental group of a closed, hyperbolic 3-manifold does not contain a subgroup isomorphic to Z 2 . This gives a profinite characterization of hyperbolicity among irreducible 3-manifolds. We also characterize Seifert fibred 3-manifolds as precisely those for which the profinite completion of the fundamental group has a non-trivial procyclic normal subgroup. Our techniques also apply to hyperbolic, virtually special groups, in the sense of Haglund and Wise. Finally, we prove that every finitely generated pro-p subgroup of the profinite completion of a torsion-free, hyperbolic, virtually special group is free pro-p.In a heuristic commonly used to distinguish 3-manifolds M and N, one computes all covers M 1 , . . . , M m of M and N 1 , . . . , N n of N up to some small degree, and then compares the resulting finite lists (M i ) and (N j ). If they can be distinguished, then one has a proof that M and N were not homeomorphic.It is natural to ask whether this method always works. A more precise question was formulated by Calegari-Freedman-Walker in [CFW10], who asked whether the fundamental group of a 3-manifold is determined by its finite quotients, or equivalently, by its profinite completion. (A standard argument shows that two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. In fact, the Geometrization theorem is equivalent to the Hyperbolization theorem and the Seifert conjecture, together with the Elliptization theorem, which asserts that M is spherical if and only if π 1 M is finite [Sco83]. Since 3-manifold groups are residually finite [Hem87], π 1 M is finite if and only if its profinite completion is, so there is no distinct profinite analogue of the Elliptization theorem. Theorems A and B can therefore be thought of as providing 2 a complete profinite analogue of the Geometrization theorem (although one should note, of course, that our proofs rely essentially on Geometrization). In Theorem 8.4, we proceed to show that the profinite completion of the fundamental group detects the geometry of a closed, orientable, irreducible 3-manifold.Alternatively, Theorem A can be thought of as a classification result for the abelian subgroups of profinite completions of fundamental groups of hyperbolic manifolds. In fact, we prove a much more general 'Tits alternative' for profinite completions of 3-manifold groups. We use the notation Z π to denote p∈π Z p .Theorem C. If M is any compact 3-manifold and H is a closed subgroup of π 1 M that does not contain a free non-abelian pro-p subgroup for any prime p then H is on the following list:1. H is conjugate to the completion of a virtually soluble subgroup of π 1 M; or, where π and σ a...

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“…If G is a hyperbolic 3-manifold group, then Agol's Virtual Fibering theorem in [AGM12] says that G has a subgroup of finite index which fibers over the circle. Hence the result of Section 3 implies that such a subgroup is COL 2 .…”

Section: Proofmentioning

confidence: 99%

Fuchsian groups, circularly ordered groups and dense invariant laminations on the circle

Baik¹

2015

Geom. Topol.

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We propose a program to study groups acting faithfully on S 1 in terms of number of pairwise transverse dense invariant laminations. We give some examples of groups which admit a small number of invariant laminations as an introduction to such groups. Main focus of the present paper is to characterize Fuchsian groups in this scheme. We prove a group acting on S 1 is conjugate to a Fuchsian group if and only if it admits three very-full laminations with a variation of the transversality condition. Some partial results toward a similar characterization of hyperbolic 3-manifold groups which fiber over the circle have been obtained. This work was motivated by the universal circle theory for tautly foliated 3-manifolds developed by Thurston and Calegari-Dunfield.

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Residual finiteness, QCERF and fillings of hyperbolic groups

Cited by 41 publications

References 18 publications

Combination of quasiconvex subgroups of relatively hyperbolic groups

Combination of quasiconvex subgroups of relatively hyperbolic groups

Distinguishing geometries using finite quotients

Fuchsian groups, circularly ordered groups and dense invariant laminations on the circle

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