Foldings, Graphs of Groups and the Membership Problem

Kapovich, Ilya; Weidmann, Richard; Myasnikov, Alexei

doi:10.1142/s021819670500213x

Cited by 77 publications

(128 citation statements)

References 34 publications

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“…Classes of groups that admit a uniform solution to the finite presentation problem include abelian groups, free groups, coherent right-angled Artin groups [27], locally quasiconvex hyperbolic groups, hyperbolic 3-manifold groups, certain Coxeter groups [37], and finitely presented residually free groups [9]. In Section 8 we shall discuss such positive results.…”

Section: Consider the Following Statement: There Exist Finitely Presementioning

confidence: 99%

On the difficulty of presenting finitely presentable groups

Bridson¹,

Wilton²

2011

Groups Geom. Dyn.

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Abstract. We exhibit classes of groups in which the word problem is uniformly solvable but in which there is no algorithm that can compute finite presentations for finitely presentable subgroups. Direct products of hyperbolic groups, groups of integer matrices, and right-angled Coxeter groups form such classes. We discuss related classes of groups in which there does exist an algorithm to compute finite presentations for finitely presentable subgroups. We also construct a finitely presented group that has a polynomial Dehn function but in which there is no algorithm to compute the first Betti number of its finitely presentable subgroups.

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Section: Consider the Following Statement: There Exist Finitely Presementioning

confidence: 99%

On the difficulty of presenting finitely presentable groups

Bridson¹,

Wilton²

2011

Groups Geom. Dyn.

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“…Morphisms of graphs of groups were introduced by Bass [2], we will use the related notion of A-graphs as presented in [18] which in turn a slight modification of the language developed in [10], in fact we assume complete familiarity with the first chapted of [18] which in particular contains a detailed description of folds as introduced by Bestvina and Feighn [3] in the language of A-graphs.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Meridional rank of knots whose exterior is a graph manifold

Boileau

Dutra

Jang

et al. 2017

Topology and its Applications

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“…Solvability of the membership problem in fundamental groups of graphs of groups was addressed by Kapovich, Miasnikov and Weidmann [51], who gave the following definition. See [7, Section 1.2] or [105] for the definition of a graph of groups.…”

Section: Theorem 411mentioning

confidence: 99%

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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Foldings, Graphs of Groups and the Membership Problem

Cited by 77 publications

References 34 publications

On the difficulty of presenting finitely presentable groups

On the difficulty of presenting finitely presentable groups

Meridional rank of knots whose exterior is a graph manifold

Decision problems for 3-manifolds and their fundamental groups

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