Rational Subgroups of Biautomatic Groups

Gersten, S. M.; Short, Hamish

doi:10.2307/2944334

Cited by 152 publications

(159 citation statements)

References 21 publications

(11 reference statements)

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“…We close this section with a discussion of the following: In an automatic or biautomatic group, a "rational subgroup" is a subgroup with a particularly simple relation to the (bi)automatic structure. See [ECH + ] for the definition; see also [GS2] for a detailed study of rational subgroups of biautomatic groups. The key fact that we propose using is: Theorems 3.3.4 and 8.3.1] If G is an automatic group and H is a rational subgroup then H is undistorted in G.…”

Section: Theorem 107 If the Artin Group A Is Right Angled Or If Itmentioning

confidence: 99%

Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

2008

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Abstract. We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single peripheral subgroup. This implies that a group being relatively hyperbolic with nonrelatively hyperbolic peripheral subgroups is a quasi-isometry invariant. As an application, Artin groups are relatively hyperbolic if and only if freely decomposable.We also introduce a new quasi-isometry invariant of metric spaces called metrically thick, which is sufficient for a metric space to be nonhyperbolic relative to any nontrivial collection of subsets. Thick finitely generated groups include: mapping class groups of most surfaces; outer automorphism groups of most free groups; certain Artin groups; and others. Nonuniform lattices in higher rank semisimple Lie groups are thick and hence nonrelatively hyperbolic, in contrast with rank one which provided the motivating examples of relatively hyperbolic groups. Mapping class groups are the first examples of nonrelatively hyperbolic groups having cut points in any asymptotic cone, resolving several questions of Drutu and Sapir about the structure of relatively hyperbolic groups. Outside of group theory, Teichmüller spaces for surfaces of sufficiently large complexity are thick with respect to the Weil-Peterson metric, in contrast with Brock-Farb's hyperbolicity result in low complexity.

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Section: Theorem 107 If the Artin Group A Is Right Angled Or If Itmentioning

confidence: 99%

Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

2008

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“…(1) if X is a compact space that is negatively curved in the sense of Alexandrov [6], then π 1 X is torsion-free and hyperbolic in the sense of Gromov [13]; (2) hyperbolic groups are biautomatic [11]; (3) the Cartesian product of two negatively (or non-positively) curved spaces is non-positively curved [6]; (4) the direct product of finitely many biautomatic groups is biautomatic [11]; (5) the fundamental groups of many (but not all) compact non-positively curved complexes are known to be biautomatic (see [12], [16]); (6) biautomatic groups are semihyperbolic [1]; (7) semihyperbolic groups are bicombable [22]. This paper is organised as follows: In the first section we consider Peiffer sequences and the second homotopy module of a presentation, prior to proving the 1-2-3 theorem in section 2.…”

Section: -2-3 Theorem Suppose Thatmentioning

confidence: 99%

“…In contrast, solvability of the conjugacy problem is not inherited by finitely pre-sented subgroups in general (see [18] -for an example of such a finite index subgroup see [8]). Thus, although products of biautomatic and related groups have a solvable conjugacy problem [11], it is possible that their finitely presented subgroups may not. This opens up the possibility of using the (un)solvability of the conjugacy problem as an invariant for identifying finitely presented subgroups of various classes of biautomatic groups that are not themselves biautomatic (or more generally bicombable [1], [21]).…”

Section: The Proof Of Theorem Amentioning

confidence: 99%

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Fibre products, non-positive curvature, and decision problems

Baumslag

Bridson²,

Miller

et al. 2000

Commentarii Mathematici Helvetici

107

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Abstract. We give a criterion for fibre products to be finitely presented and use it as the basis of a construction that encodes the pathologies of finite group presentations into pairs of groups P ⊂ G where G is a product of hyperbolic groups and P is a finitely presented subgroup. This enables us to prove that there is a finitely presented subgroup P in a biautomatic group G such that the generalized word problem for P ⊂ G is unsolvable and P has an unsolvable conjugacy problem. An additional construction shows that there exists a compact non-positively curved polyhedron X such that π 1 X is biautomatic and there is no algorithm to decide isomorphism among the finitely presented subgroups of π 1 X.

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“…The positive part of this statement ultimately derives from the fact that translation numbers τ (γ) = lim d(1, γ n )/n are positive for elements of infinite order, while the negative part derives from the fact that, unlike in Hyp and C 0 , one does not know if the set of these numbers is discrete. These results are proved in [2] and [90] using ideas from [59].…”

Section: Non-positively Curved Groupsmentioning

confidence: 78%

Non-positive curvature and complexity for finitely presented groups

Bridson¹

2007

Proceedings of the International Congress of Mathematicians Madrid, August 22–30, 2006

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Abstract. A universe of finitely presented groups is sketched and explained, leading to a discussion of the fundamental role that manifestations of non-positive curvature play in group theory. The geometry of the word problem and associated filling invariants are discussed. The subgroup structure of direct products of hyperbolic groups is analysed and a process for encoding diverse phenomena into finitely presented subdirect products is explained. Such an encoding is used to solve problems of Grothendieck concerning profinite completions and representations of groups. In each context, explicit groups are crafted to solve problems of a geometric nature.

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Rational Subgroups of Biautomatic Groups

Cited by 152 publications

References 21 publications

Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity

Fibre products, non-positive curvature, and decision problems

Non-positive curvature and complexity for finitely presented groups

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