The conjugacy and isomorphism problems for combable groups

Bridson, Martin R.

doi:10.1007/s00208-003-0453-6

Cited by 5 publications

(4 citation statements)

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“…There are several natural classes of finitely presented groups that cluster around the notion of non-positive curvature, ranging from hyperbolic groups to combable groups (see [6] for a survey and references). The isomorphism problem is solvable in the class of hyperbolic groups but is unsolvable in the class of combable groups [4]. It remains unknown whether the isomorphism is solvable in the intermediate classes, such as (bi)automatic groups and CAT(0) groups.…”

Section: For Fabrizio Catanese On His 60th Birthdaymentioning

confidence: 99%

On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

Bridson

Reeves

2011

Sci. China Math.

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We show that the isomorphism problem is solvable in the class of central extensions of wordhyperbolic groups, and that the isomorphism problem for biautomatic groups reduces to that for biautomatic groups with finite centre. We describe an algorithm that, given an arbitrary finite presentation of an automatic group Γ, will construct explicit finite models for the skeleta of K(Γ, 1) and hence compute the integral homology and cohomology of Γ. Keywordsgeometric group theory, automatic groups, isomorphism problem MSC(2000): 20F65, 20F67, 57M07 Citation: Bridson M R, Reeves L. On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups.There are several natural classes of finitely presented groups that cluster around the notion of non-positive curvature, ranging from hyperbolic groups to combable groups (see [6] for a survey and references). The isomorphism problem is solvable in the class of hyperbolic groups but is unsolvable in the class of combable groups [4]. It remains unknown whether the isomorphism is solvable in the intermediate classes, such as (bi)automatic groups and CAT(0) groups. Hyperbolic groups also form one of the very few interesting classes in which there is an algorithm that, given a finite presentation of a group Γ in the class, will construct finite models for the skeleta of a K(Γ, 1). For finitely presented groups in general, one cannot even calculate H 2 (Γ, Z); see [13]. Our focus in this article will be on the isomorphism problem for biautomatic groups and the construction problem for classifying spaces of combable and automatic groups.We remind the reader that the isomorphism problem for a class G of finitely presented groups is said to be solvable if there exists an algorithm that takes as input pairs of finite group presentations (P 1 , P 2 ) and proceeding under the assumption that the groups |P i | belong to C, decides whether or not |P 1 | ∼ = |P 2 |. The first purpose of this article is to point out that the isomorphism problem for biautomatic groups (or any subclass of such groups) can be reduced to the problem of determining isomorphism of the groups modulo their centres. We write Z(G) to denote the centre of a group G.

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Section: For Fabrizio Catanese On His 60th Birthdaymentioning

confidence: 99%

On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

Bridson

Reeves

2011

Sci. China Math.

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“…This control carries over to SH and bicombable groups, where it yields a solution to the conjugacy problem. But control is lost as one weakens the link to CAT(0) spaces [27]. Theorem 3.3.…”

Section: The Conjugacy and Isomorphism Problems If Two Rectifiablementioning

confidence: 99%

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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“…The issue here is one of Nielsen equivalence between different generating sets of im ψ. It is to circumvent this difficulty that we took F above to be a free group of rank 2m rather than simply m. This apparent redundancy has been employed so that we can appeal to the following version of Rapaport's Theorem [13], which is proved is Section 4 of [6]. …”

Section: A Seed Of Undecidabilitymentioning

confidence: 99%

Recognition of subgroups of direct products of hyperbolic groups

Bridson¹,

Miller²

2003

Proc. Amer. Math. Soc.

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Abstract. We give examples of direct products of three hyperbolic groups in which there cannot exist an algorithm to decide which finitely presented subgroups are isomorphic.In [1] Baumslag, Short and the present authors constructed an example of a biautomatic group G in which there is no algorithm that decides isomorphism among the finitely presented subgroups of G. The group G was a direct product of a certain (word) hyperbolic group H with an HNN extension B of H; the group B was CAT(0) and biautomatic but not hyperbolic. At the time of writing [1] we were unable to construct, more simply, a direct product of hyperbolic groups in which the isomorphism problem for finitely presented subgroups was unsolvable.In the course of our project on finiteness properties of subdirect products [7], we uncovered a new criterion for the finite presentability of certain semidirect products (see Section 2). That, combined with an improved understanding of subgroups of direct products, has enabled us to prove the following result: An example of such a group Γ is given, for instance, by applying the construction of Rips [14] to a non-abelian free group.

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The conjugacy and isomorphism problems for combable groups

Abstract: There exist combable groups in which the conjugacy problem is unsolvable. The isomorphism problem is unsolvable for certain recursive sequences of finite presentations of combable groups.

Cited by 5 publications

References 20 publications

On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

On the algorithmic construction of classifying spaces and the isomorphism problem for biautomatic groups

Non-positive curvature and complexity for finitely presented groups

Recognition of subgroups of direct products of hyperbolic groups

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