Describing free groups

Carson, Jacob; Harizanov, Valentina S.; Knight, Julia F.; Lange, Karen; McCoy, Charles F. D.; Morozov, Andrey; Quinn, S.; Safranski, C.; Wallbaum, John

doi:10.1090/s0002-9947-2012-05456-0

Cited by 23 publications

(30 citation statements)

References 13 publications

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“…For purposes of characterizing complexity, we will use the following framework (note that this definition is equivalent to that given in [6]). Definition 1.…”

Section: Definitionsmentioning

confidence: 99%

Detecting properties from descriptions of groups

2019

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We consider whether given a simple, finite description of a group in the form of an algorithm, it is possible to algorithmically determine if the corresponding group has some specified property or not. When there is such an algorithm, we say the property is recursively recognizable within some class of descriptions. When there is not, we ask how difficult it is to detect the property in an algorithmic sense.We consider descriptions of two sorts: first, recursive presentations in terms of generators and relators, and second, algorithms for computing the group operation. For both classes of descriptions, we show that a large class of natural algebraic properties, Markov properties, are not recursively recognizable, indeed they are Π 0 2 -hard to detect in recursively presented groups and Π 0 1 -hard to detect in computable groups. These theorems suffice to give a sharp complexity measure for the detection problem of a number of typical group properties, for example, being abelian, torsion-free, orderable. Some properties, like being cyclic, nilpotent, or solvable, are much harder to detect, and we give sharp characterizations of the corresponding detection problems from a number of them.We give special attention to orderability properties, as this was a main motivation at the beginning of this project.

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“…For purposes of characterizing complexity, we will use the following framework (note that this definition is equivalent to that given in [6]). Definition 1.…”

Section: Definitionsmentioning

confidence: 99%

Detecting properties from descriptions of groups

2019

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“…As a consequence of these technical difficulties, our knowledge of Δ 0 n -categorical structures is rather limited even when n = 2. Only recently, there has been significant progress in understanding Δ 0 n -categoricity in several specific classes, for small n. It follows from [8,30] that every free (non-abelian) group of rank ω is Δ 0 3 -categorical, and the result cannot be improved to Δ 0 2 . It is also known that every computable completely decomposable group is Δ 0 5 -categorical, and the result is sharp [13].…”

Section: Non-computable Isomorphisms Between Computable Structuresmentioning

confidence: 99%

OnΔ20-categoricity of equivalence relations

Downey¹,

Melnikov²,

Ng³

2015

Annals of Pure and Applied Logic

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“…In fact, the computable finitely generated groups that we have studied all have Scott sentences that are "computable d-Σ2" (the conjunction of a computable Σ2 sentence and a computable Π2 sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group.…”

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confidence: 92%

“…Hence, there can be no simpler Scott sentence. In [9], it is shown that the free group F n on n generators has a Scott sentence that is computable d-Σ 2 . It is also shown that I(F n ) is m-complete d-Σ 0 2 , so there is no simpler Scott sentence.…”

Section: Introductionmentioning

confidence: 99%

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Scott sentences for certain groups

Knight

Saraph

2017

Arch. Math. Logic

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We give Scott sentences for certain computable groups, and we use index set calculations as a way of checking that our Scott sentences are as simple as possible. We consider finitely generated groups and torsion-free abelian groups of finite rank. For both kinds of groups, the computable ones all have computable Σ3 Scott sentences. Sometimes we can do better. In fact, the computable finitely generated groups that we have studied all have Scott sentences that are "computable d-Σ2" (the conjunction of a computable Σ2 sentence and a computable Π2 sentence). In [9], this was shown for the finitely generated free groups. Here we show it for all finitely generated abelian groups, and for the infinite dihedral group. 1 Among the computable torsion-free abelian groups of finite rank, we focus on those of rank 1. These are exactly the additive subgroups of Q. We show that for some of these groups, the computable Σ3 Scott sentence is best possible, while for others, there is a computable d-Σ2 Scott sentence.

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Describing free groups

Cited by 23 publications

References 13 publications

Detecting properties from descriptions of groups

Detecting properties from descriptions of groups

OnΔ20-categoricity of equivalence relations

Scott sentences for certain groups

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