Determining Fuchsian groups by their finite quotients

Bridson, Martin R.; Conder, Marston; Reid, Alan W.

doi:10.1007/s11856-016-1341-6

Cited by 39 publications

(77 citation statements)

References 57 publications

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“…(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.) Bridson, Conder and Reid have answered the corresponding question for Fuchsian groups positively [BCR14], while Long and Reid have given a positive answer to a related question [LR11]. We refer the reader to Section 8 of [Rei13] for a discussion of this and related problems.…”

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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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confidence: 99%

Distinguishing geometries using finite quotients

Wilton

Zalesskiĭ

2017

Geom. Topol.

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“…We note that recently Bridson, Conder and Reid [6] have proved that Δ(l, m, n) ∼ = Δ(l , m , n ) if and only if Δ(l, m, n) ∼ = Δ(l , m , n ).…”

Section: Grothendieck's Theory With Typesmentioning

confidence: 73%

The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces

González-Diez

Jaikin–Zapirain

2015

Proc. London Math. Soc.

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Beauville surfaces are an important kind of algebraic surfaces introduced by Catanese. They are rigid surfaces of general type defined over number fields. We prove that, for any σ∈Gal(double-struckQ¯/Q) different from the identity and the complex conjugation, there is a Beauville surface S such that S and its Galois conjugate Sσ have non‐isomorphic fundamental groups. This in turn easily implies that the action of Gal(double-struckQ¯/Q) on the set of isomorphism classes of Beauville surfaces is faithful. These results were conjectured by Bauer, Catanese and Grunewald, and immediately imply that Gal(double-struckQ¯/Q) acts faithfully on the connected components of the moduli space of surfaces of general type, a result due to the above‐mentioned authors. These results on Beauville surfaces heavily depend on the fact that the absolute Galois group acts faithfully on regular dessins, a result that we prove in this paper. Moreover, we show the stronger result that the action is faithful on the set of quasiplatonic (or triangle) curves of any given hyperbolic type.

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“…Indeed, limit groups are closely related to free groups (for instance, Remeslen-nikov showed that they are precisely the existentially free groups [34]), and are frequently hard to distinguish from them. Bridson, Conder and Reid [6] pointed out that Corollary C, combined with the results of [41], would resolve Remeslennikov's question in this case.…”

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confidence: 90%

Essential surfaces in graph pairs

Wilton¹

2018

J. Amer. Math. Soc.

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A well known question of Gromov asks whether every one-ended hyperbolic group Γ has a surface subgroup. We give a positive answer when Γ is the fundamental group of a graph of free groups with cyclic edge groups. As a result, Gromov's question is reduced (modulo a technical assumption on 2-torsion) to the case when Γ is rigid. We also find surface subgroups in limit groups. It follows that a limit group with the same profinite completion as a free group must in fact be free, which answers a question of Remeslennikov in this case.

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Determining Fuchsian groups by their finite quotients

Cited by 39 publications

References 57 publications

Distinguishing geometries using finite quotients

Distinguishing geometries using finite quotients

The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces

Essential surfaces in graph pairs

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