Equation problem over central extensions of hyperbolic groups

Liang, Hao

doi:10.1142/s1793525314500095

Cited by 3 publications

(3 citation statements)

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“…2-generator) Artin groups, and groups that are virtually certain types of right-angled Artin groups. We note that, since any dihedral Artin group can alternatively be decomposed as a central extension of Z by a virtually free group, decidability of its systems of equations (but not the more general problem with recognisable constraints) could also be derived from [16].…”

Section: Introductionmentioning

confidence: 99%

Equations in groups that are virtually direct products

2020

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Section: Introductionmentioning

confidence: 99%

Equations in groups that are virtually direct products

2020

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“…It generalizes both the word problem and the conjugacy problem, and has been solved for torsion-free hyperbolic groups by Makanin [62] and Rips and Sela [95], for hyperbolic groups with torsion by Dahmani and Guirardel [21] and for fundamental groups of Seifert fibered spaces by Liang [60]. The following question thus arises.…”

Section: Open Problemsmentioning

confidence: 99%

“…The Equation Problem asks for a solution to the problem whether any set of 'equations' over a group has a solution. It generalizes both the Word Problem and the Conjugacy Problem, and has been solved for torsion-free hyperbolic groups by Makanin [Mak82] and Rips-Sela [RS98], for hyperbolic groups with torsion by Dahmani-Guirardel [DGu10] and for fundamental groups of Seifert fibered spaces by Liang [Li14]. The following question thus arises.…”

Section: Open Problemsmentioning

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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Word Equations, Constraints, and Formal Languages

Ciobanu

2024

Lecture Notes in Computer Science

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Equation problem over central extensions of hyperbolic groups

Abstract: The Equation Problem in finitely presented groups asks if there exists an algorithm which determines in finite amount of time whether any given equation system has a solution or not. We show that the Equation Problem in central extensions of hyperbolic groups is solvable.

Cited by 3 publications

References 12 publications

Equations in groups that are virtually direct products

Equations in groups that are virtually direct products

Decision problems for 3-manifolds and their fundamental groups

Word Equations, Constraints, and Formal Languages

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