Subgroups of Surface Groups are Almost Geometric

Scott, Peter

doi:10.1112/jlms/s2-17.3.555

Cited by 326 publications

(356 citation statements)

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“…In the Seifert fibered case this was proved by Niblo [80] building on the aforementioned work of Scott [98]. In the hyperbolic case this is an extension of Theorem 3.20, which follows from work of Wise [116,Theorem 16.23] combined with work of Hruska [43,Corollary 1.6].…”

Section: Remark 415mentioning

confidence: 58%

“…As mentioned before, in this case, is subgroup separable by [98]; this immediately gives a uniform solution to the membership problem. If N is not Seifert fibered, then we use Theorem 3.14 to determine the JSJ decomposition of N .…”

Section: Remark 415mentioning

confidence: 98%

“…Using the argument of the proof of Lemma 4.3 we then obtain a solution to the uniform membership problem for fundamental groups of hyperbolic 3-manifolds. This approach works also for fundamental groups of Seifert fibered spaces, which are subgroup separable by work of Scott [98]. The argument can also be used for special classes of subgroups, eg subgroups carried by embedded surfaces [92], but this approach does not work in general.…”

Section: Theorem 411mentioning

confidence: 99%

“…†/ < 0. Note that, since every finitely generated subgroup of a surface group 1 † is a virtual retract (this is implicit in [98]), it follows easily that every finitely generated subgroup of 1 † Z is a virtual retract. Therefore, a naive search using the Reidemeister-Schreier algorithm will find a finite-index subgroup 0 of and a retraction W 0 !…”

Section: Remark 415mentioning

confidence: 99%

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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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Section: Remark 415mentioning

confidence: 58%

Section: Remark 415mentioning

confidence: 98%

Section: Theorem 411mentioning

confidence: 99%

Section: Remark 415mentioning

confidence: 99%

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Decision problems for 3-manifolds and their fundamental groups

Aschenbrenner

Friedl

Wilton

2015

Geometry &Amp; Topology Monographs

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“…Subgroup separability is not only of interest group-theoretically but also is of interest for topological reasons. Thus Scott [10], using geometrical methods, showed that orientable surface groups are subgroup separable. This result was generalized by Brunner, Burns, and Solitar [3] who showed that the generalized free products of free groups with cyclic amalgamation are subgroup separable.…”

Section: Introductionmentioning

confidence: 99%

On the subgroup separability of generalized free products of nilpotent groups

Tang¹

1991

Proc. Amer. Math. Soc.

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Regular Maps on Surfaces with Large Planar Width

Nedela

Škoviera

2001

European Journal of Combinatorics

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A map is a cell decomposition of a closed surface; it is regular if its automorphism group acts transitively on the flags, mutually incident vertex-edge-face triples. The main purpose of this paper is to establish, by elementary methods, the following result: for each positive integer w and for each pair of integers p ≥ 3 and q ≥ 3 satisfying 1/ p + 1/q ≤ 1/2, there is an orientable regular map with face-size p and valency q such that every non-contractible simple closed curve on the surface meets the 1-skeleton of the map in at least w points. This result has several interesting consequences concerning maps on surfaces, graphs and related concepts. For example, MacBeath's theorem about the existence of infinitely many Hurwitz groups, or Vince's theorem about regular maps of given type ( p, q), or residual finiteness of triangle groups, all follow from our result.

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Subgroups of Surface Groups are Almost Geometric

Cited by 326 publications

References 5 publications

Decision problems for 3-manifolds and their fundamental groups

Decision problems for 3-manifolds and their fundamental groups

On the subgroup separability of generalized free products of nilpotent groups

Regular Maps on Surfaces with Large Planar Width

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