Géométrie et théorie des groupes

Coornaert, Michel; Delzant, Thomas; Papadopoulos, Athanase

doi:10.1007/bfb0084913

Cited by 294 publications

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“…By [12], a group G is hyperbolic if and only if G ∼ = Γ * /R, where R is finite, length-reducing, and {s ∈ Γ * | s *…”

Section: Theorem 2 the Word Problem For Every Fixed Hyperbolic Group mentioning

confidence: 99%

Decidability and Complexity in Automatic Monoids

Lohrey

2004

Developments in Language Theory

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Several complexity and decidability results for automatic monoids are shown: (i) there exists an automatic monoid with a P-complete word problem, (ii) there exists an automatic monoid such that the first-order theory of the corresponding Cayley-graph is not elementary decidable, and (iii) there exists an automatic monoid such that reachability in the corresponding Cayley-graph is undecidable. Moreover, it is shown that for every hyperbolic group the word problem belongs to LOGCFL, which improves a result of Cai [8].Keywords: automatic monoids, hyperbolic groups, word problems, Cayley-graphs, complexity, decidability IntroductionAutomatic groups attracted a lot of attention in combinatorial group theory during the last 15 years, see e.g. the textbook [15]. Roughly speaking, a finitely generated group G, generated by the finite set Γ, is automatic, if the elements of G can be represented by words from a regular language over Γ, and the multiplication with a generator can be recognized by a synchronized 2-tape automaton. This concept easily yields a quadratic time algorithm for the word problem of an automatic group.It is straight forward to extend the definition of an automatic group to the monoid case; this leads to the class of automatic monoids, see e.g. [10,18,22,36]. In the present paper, we study the complexity and decidability of basic algorithmic questions in automatic monoids. In Section 4 we consider the complexity of the word problem for automatic monoids. Analogously to the group case, it is easy to show that for every automatic monoid the word problem can be solved in quadratic time. Here, we prove that there exists a fixed automatic monoid with a P-complete word problem. Thus, unless P = NC, where NC is the class of all problems that can be solved in polylogarithmic time using a polynomial amount of hardware, there exist * This work was partly done while the author was at RWTH Aachen, Germany. 1 automatic monoids for which the word problem cannot be efficiently parallelized. Whether there exists an automatic group with a P-complete word problem was asked for the first time by Cai [8]. This problem remains open.An important subclass of the class of automatic groups is the class of hyperbolic groups, which are defined via a geometric hyperbolicity condition on the Cayleygraph [16]. In [8], Cai has shown that for every hyperbolic group the word problem belongs to the parallel complexity class NC 2 . Cai also asked, whether the upper bound of NC 2 can be improved. Using known results from formal language theory, we show in Section 4 that the word problem for every hyperbolic group belongs to the complexity class LOGCFL ⊆ NC 2 . LOGCFL is the class of all problems that are logspace reducible to a context-free language [44]. We also present a class of automatic monoids, namely monoids that can be presented by finite, terminating, confluent, and left-basic semi-Thue systems [41], for which the complexity of the word problem captures the class LOGDCFL (the logspace closure of the deterministic contex...

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“…By [12], a group G is hyperbolic if and only if G ∼ = Γ * /R, where R is finite, length-reducing, and {s ∈ Γ * | s *…”

Section: Theorem 2 the Word Problem For Every Fixed Hyperbolic Group mentioning

confidence: 99%

Decidability and Complexity in Automatic Monoids

Lohrey

2004

Developments in Language Theory

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“…Hence a simply connected surface is open if and only if it is topologically equivalent to the Euclidean plane. Theorem 3.3 was proved in [4,Chapter 6] when the surface is a two dimensional complete simply connected Riemannian manifolds, and then it was slightly extended in [11] to general simply connected Aleksandrov surfaces, even without the assumption about completeness.…”

Section: Theorem 33 (Gromov) Every Open Simply Connected Aleksandromentioning

confidence: 99%

Hyperbolic Notions on a Planar Graph of Bounded Face Degree

Oh¹

2015

Bulletin of the Korean Mathematical Society

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Abstract. We study the relations between strong isoperimetric inequalities and Gromov hyperbolicity on planar graphs, and give an alternative proof for the following statement: if a planar graph of bounded face degree satisfies a strong isoperimetric inequality, then it is Gromov hyperbolic. This theorem was formerly proved in the author's paper from 2014 [12] using combinatorial methods, while geometric approach is used in the present paper.

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“…Its importance is widely appreciated. Gromov hyperbolicity was introduced by Gromov in the setting of geometric group theory [32], [33], [31], [25], but has played an increasing role in analysis on general metric spaces [12], [13], [7], with applications to the Martin boundary, invariant metrics in several complex variables [6] and extendability of Lipschitz mappings [42]. Here we survey the basics of Gromov hyperbolicity.…”

Section: Gromov Hyperbolicitymentioning

confidence: 99%

“…Here we survey the basics of Gromov hyperbolicity. For detailed expositions, see for instance [25], [31], [14, II.H], or [47].…”

Section: Gromov Hyperbolicitymentioning

confidence: 99%

Nonpositive curvature and complex analysis

Buckley¹

2010

Five Lectures in Complex Analysis

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Abstract. We discuss a few of the metrics that are used in complex analysis and potential theory, including the Poincaré, Carathéodory, Kobayashi, Hilbert, and quasihyperbolic metrics. An important feature of these metrics is that they are quite often negatively curved. We discuss what this means and when it occurs, and proceed to investigate some notions of nonpositive curvature, beginning with constant negative curvature (e.g. the unit disk with the Poincaré metric), and moving on to CAT(k) and Gromov hyperbolic spaces. We pay special attention to notions of the boundary at infinity.

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Géométrie et théorie des groupes

Cited by 294 publications

References 0 publications

Decidability and Complexity in Automatic Monoids

Decidability and Complexity in Automatic Monoids

Hyperbolic Notions on a Planar Graph of Bounded Face Degree

Nonpositive curvature and complex analysis

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