Growth of Certain Non-positively Curved Cube Groups

Noskov, G. A.

doi:10.1006/eujc.1999.0372

Cited by 8 publications

(7 citation statements)

References 17 publications

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“…And conversely a group is right-angled if it admits an action on a CAT (0) cube complex X which is simply transitive on the set of vertices. So right-angled groups are the groups considered by Noskov in [29].…”

Section: Definition 231mentioning

confidence: 99%

Finite index subgroups of graph products

2008

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Section: Definition 231mentioning

confidence: 99%

Finite index subgroups of graph products

2008

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“…In particular it follows from Proposition 4.4 and Theorem 4.3 in their paper that all finitely generated abelian groups have β-stable intervals and that finitely generated virtually abelian groups as well as geometrically finite hyperbolic groups have β-stable intervals for some word metrics. The FFT property has been verified for further classes of groups, with respect to suitably chosen finite generating sets, in [33,34,22].…”

Section: U Langmentioning

confidence: 99%

Injective Hulls of Certain Discrete Metric Spaces and Groups

Lang

2013

J. Topol. Anal.

160

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Injective metric spaces, or absolute 1-Lipschitz retracts, share a number of properties with CAT(0) spaces. In the '60s Isbell showed that every metric space X has an injective hull E(X). Here it is proved that if X is the vertex set of a connected locally finite graph with a uniform stability property of intervals, then E(X) is a locally finite polyhedral complex with finitely many isometry types of n-cells, isometric to polytopes in l n ∞ , for each n. This applies to a class of finitely generated groups Γ, including all word hyperbolic groups and abelian groups, among others. Then Γ acts properly on E(Γ) by cellular isometries, and the first barycentric subdivision of E(Γ) is a model for the classifying space EΓ for proper actions. If Γ is hyperbolic, E(Γ) is finite dimensional and the action is cocompact. In particular, every hyperbolic group acts properly and cocompactly on a space of non-positive curvature in a weak (but non-coarse) sense.finite then the injective hull is a finite polyhedral complex of dimension at most 1 2 |X| whose n-cells are isometric to polytopes in l n ∞ = l ∞ ({1, . . . , n}). A detailed account of injective metric spaces and hulls is given below, in Secs. 2 and 3.Isbell's construction was rediscovered 20 years later by Dress [13] (and even another time in [10]). Due to this independent work and a characterization of injective metric spaces from [3], metric injective hulls are also called tight spans or hyperconvex hulls in the literature, furthermore "hull" is often substituted by "envelope". Tight spans are widely known in discrete mathematics and have notably been used in phylogenetic analysis (see [17,16] for some surveys). Apart from the two foundational papers [24,13] and some work referring to Banach spaces (see, for instance, [25,36,11]), the vast literature on metric injective hulls deals almost exclusively with finite metric spaces. Dress proved that for certain discrete metric spaces X the tight span T X still has a polyhedral structure (see Items (5.19), (6.2), and (6.6) in [13]); these results, however, presuppose that T X is locally finite dimensional. A simple sufficient, geometric condition on X to this effect has been missing (but see Theorem 9 and Item (5.14) in [13]).Here it is now shown that, in the case of integer valued metrics, a weak form of the fellow traveler property for discrete geodesics serves the purpose and even ensures that E(X) is proper, provided X is; see Theorem 1.1 below. The polyhedral structure of E(X) and the possible isometry types of cells are described in detail and no prior knowledge of the constructions in [24, 13] is assumed. With regard to applications in geometric group theory, a general fixed point theorem for injective metric spaces is pointed out (Proposition 1.2), which closely parallels the wellknown result for CAT(0) spaces. It has been known for some time that if the metric space X is δ-hyperbolic, then so is E(X), and this implies that E(X) is within finite distance of e(X), provided X is geodesic or discretely geodesic (Propos...

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“…Families of groups that have the falsification by fellow traveler property with respect to some generating set include virtually abelian groups [27], geometrically finite hyperbolic groups [27], Coxeter groups and groups acting simply transitively on the chambers of locally finite buildings [31], groups acting cellularly on locally finite CAT(0) cube complexes where the action is simply transitive on the vertices [30], Garside groups [21] and Artin groups of large type [22].…”

Section: Introductionmentioning

confidence: 99%

Finite generating sets of relatively hyperbolic groups and applications to geodesic languages

Antolín

Ciobanu

2016

Trans. Amer. Math. Soc.

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Abstract. Given a finitely generated relatively hyperbolic group G, we construct a finite generating set X of G such that (G, X) has the 'falsification by fellow traveler property' provided that the parabolic subgroups {Hω}ω∈Ω have this property with respect to the generating sets {X ∩ Hω}ω∈Ω. This implies that groups hyperbolic relative to virtually abelian subgroups, which include all limit groups and groups acting freely on R n -trees, or geometrically finite hyperbolic groups, have generating sets for which the language of geodesics is regular, and the complete growth series and complete geodesic series are rational. As an application of our techniques, we prove that if each Hω admits a geodesic biautomatic structure over X ∩ Hω, then G has a geodesic biautomatic structure.Similarly, we construct a finite generating set X of G such that (G, X) has the 'bounded conjugacy diagrams' property or the 'neighbouring shorter conjugate' property if the parabolic subgroups {Hω}ω∈Ω have this property with respect to the generating sets {X ∩Hω}ω∈Ω. This implies that a group hyperbolic relative to abelian subgroups has a generating set for which its Cayley graph has bounded conjugacy diagrams, a fact we use to give a cubic time algorithm to solve the conjugacy problem. Another corollary of our results is that groups hyperbolic relative to virtually abelian subgroups have a regular language of conjugacy geodesics.

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Growth of Certain Non-positively Curved Cube Groups

Cited by 8 publications

References 17 publications

Finite index subgroups of graph products

Finite index subgroups of graph products

Injective Hulls of Certain Discrete Metric Spaces and Groups

Finite generating sets of relatively hyperbolic groups and applications to geodesic languages

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