Two Automatic Spanning Trees in Small Cancellation Group Presentations

Johnsgard, Karin

doi:10.1142/s0218196796000258

Cited by 2 publications

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“…Gilman has given other characterizations of groups that are automatic with respect to a prefix-closed normal form set in [12]. Groups known to have an automatic structure with respect to prefix-closed normal forms include finite groups [10], virtually abelian (and hence Euclidean) groups and word hyperbolic groups [10], Coxeter groups [5], Artin groups of finite type [7] and of large type [28], [19], and small cancellation groups satisfying conditions C ′′ (p) − T (q) for (p, q) ∈ { (3,6), (4,4), (6,3)} [21]. The class of automatic groups with respect to prefix-closed normal forms is closed under graph products [15,Theorem B] and finite extensions [10,Theorem 4.1.4].…”

Section: Introductionmentioning

confidence: 99%

Algorithms and topology of Cayley graphs for groups

2014

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Autostackability for finitely generated groups is defined via a topological property of the associated Cayley graph which can be encoded in a finite state automaton. Autostackable groups have solvable word problem and an effective inductive procedure for constructing van Kampen diagrams with respect to a canonical finite presentation. A comparison with automatic groups is given. Another characterization of autostackability is given in terms of prefix-rewriting systems. Every group which admits a finite complete rewriting system or an asynchronously automatic structure with respect to a prefix-closed set of normal forms is also autostackable. As a consequence, the fundamental group of every closed 3-manifold with any of the eight possible uniform geometries is autostackable.

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Section: Introductionmentioning

confidence: 99%

Algorithms and topology of Cayley graphs for groups

2014

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Small cancellation groups and translation numbers

Kapovich

1997

Trans. Amer. Math. Soc.

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Abstract. In this paper we prove that C(4)-T(4)-P, C(3)-T(6)-P and C(6)-P small cancellation groups are translation discrete in the strongest possible sense and that in these groups for any g and any n there is an algorithm deciding whether or not the equation x n = g has a solution. There is also an algorithm for calculating for each g the maximum n such that g is an n-th power of some element. We also note that these groups cannot contain isomorphic copies of the group of p-adic fractions and so in particular of the group of rational numbers. Besides we show that for C (4) − T (4) and C (3) − T (6) groups all translation numbers are rational and have bounded denominators. IntroductionIn [GS1] S.Gersten and H.Short introduced the notion of translation numbers for finitely generated groups. This concept was inspired by some considerations about groups acting on R-trees and about the action of the fundamental group of a closed riemannian manifold of nonpositive curvature on the universal cover of this manifold (see, for example, the book of W.Ballmann, M.Gromov and V.Schroeder, "Manifolds of nonpositive curvature" [BGrS]).Definition. Let G be a group and let X be a finite generating set of G closed under inversion. Then any element g in G can be expressed as a product g = x 1 ·x 2 ·· · ··x n where x i ∈ X; we call the minimal such n the X-length of g and denote it l X (g).Then for any g ∈ G we define the translation number of g with respect to X as follows:(It is noted in [GS1] that this limit always exists.)From now on we will assume that all generating sets for all groups in this section are closed under inversions. It turns out (see [GS1] for proofs) that if G is a group and X is a finite generating set, then:(a) τ X (g) = τ X (g −1 ) ≤ l X (g); (b) τ X (g) = τ X (hgh −1 ); (c) if g is an element of finite order then τ X (g) = 0; (d) τ X (g n ) = nτ X (g) for any integer n and for any g ∈ G;

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Two Automatic Spanning Trees in Small Cancellation Group Presentations

Cited by 2 publications

References 0 publications

Algorithms and topology of Cayley graphs for groups

Algorithms and topology of Cayley graphs for groups

Small cancellation groups and translation numbers

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