A topological zero-one law and elementary equivalence of finitely generated groups

Osin, Denis

doi:10.1016/j.apal.2020.102915

Cited by 9 publications

(9 citation statements)

References 50 publications

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“…We show that, in general, elementary equivalence does not impose any restrictions on quasi‐isometric diversity. The proof is a simple combination of Corollary 1.2 and a recent result of the second author [34, Theorem 2.9], stating that

{\bar{scriptH}}_{0}

contains a comeagre elementary equivalence class. Corollary There exist

2^{ℵ_{0}}

finitely generated, pairwise non‐quasi‐isometric, elementarily equivalent groups.…”

Section: Introductionmentioning

confidence: 94%

“…The relevance of the infinitary logic to the study of generic properties is demonstrated by the proposition below, which has long been known to experts. The proof is fairly elementary and can be found in [34, Proposition 5.1] (see also [19, Proposition 3.2]), but we provide a sketch for convenience of the reader. Proposition A

Π_{2}

‐definable group‐theoretic property is a

G_{δ}

‐property.…”

Section: Generic Properties and Limits Of Hyperbolic Groupsmentioning

confidence: 99%

“…The set

{\bar{scriptH}}_{0}

is known to be perfect (see, for example, [34, Theorem 2.9]). Since every hyperbolic group is finitely presented, Corollary 1.2 can be applied to

S = {\bar{H}}_{0}

.…”

Section: Introductionmentioning

confidence: 99%

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Quasi‐isometric diversity of marked groups

Minasyan

Osin

Witzel³

2021

Journal of Topology

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We use basic tools of descriptive set theory to prove that a closed set S of marked groups has 2 ℵ 0 quasi-isometry classes, provided that every non-empty open subset of S contains at least two non-quasi-isometric groups. It follows that every perfect set of marked groups having a dense subset of finitely presented groups contains 2 ℵ 0 quasi-isometry classes. These results account for most known constructions of continuous families of non-quasi-isometric finitely generated groups. We use them to prove the existence of 2 ℵ 0 quasi-isometry classes of finitely generated groups having interesting algebraic, geometric, or model-theoretic properties (for example, such groups can be torsion, simple, verbally complete or they can all have the same elementary theory).

show abstract

{\bar{scriptH}}_{0}

contains a comeagre elementary equivalence class. Corollary There exist

2^{ℵ_{0}}

finitely generated, pairwise non‐quasi‐isometric, elementarily equivalent groups.…”

Section: Introductionmentioning

confidence: 94%

Π_{2}

‐definable group‐theoretic property is a

G_{δ}

‐property.…”

Section: Generic Properties and Limits Of Hyperbolic Groupsmentioning

confidence: 99%

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Quasi‐isometric diversity of marked groups

Minasyan

Osin

Witzel³

2021

Journal of Topology

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“…8) Recall the following result [53], [42,Theorem 3.5] : For every finitely generated group G, there exists an L ω1,ω -sentence σ with the property that for any group H we have…”

Section: Properties Of Subgroupsmentioning

confidence: 99%

Rich groups, weak second order logic, and applications

Kharlampovich¹,

Myasnikov²,

Sohrabi³

2021

Preprint

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In this paper we initiate a study of first-order rich groups, i.e., groups where the first-order logic has the same power as the weak second order logic. Surprisingly, there are quite a lot of finitely generated rich groups, they are somewhere in between hyperbolic and nilpotent groups (these ones are not rich). We provide some methods to prove that groups (and other structures) are rich and describe some of their properties. As corollaries we look at Malcev's problems in various groups.

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“…The topology on the space of marked groups can be traced back to Chabauty in [Cha50]. It has by now become a standard tool in the study of group properties, see for instance [MOW19] and [Osi21] where tools of descriptive set theory are used to obtain far reaching group theoretical results. We will still include a brief introduction to describe the topology and the metric of the space of marked groups, which we denote by G throughout.…”

Section: Introductionmentioning

confidence: 99%

Computable analysis on the space of marked groups

Rauzy¹

2021

Preprint

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We investigate decision problems for groups described by word problem algorithms. We show that this corresponds to the study of computable analysis on the space of marked groups, and point out several results of computable analysis that can be directly applied to obtain group theoretical results. However, we prove that the space of marked groups is a Polish space that is not effectively Polish, because of this, many of the most important results of computable analysis cannot be applied to the space of marked groups. This includes the Kreisel-Lacombe-Schoenfield-Ceitin Theorem and different theorems of Moschovakis. The space of marked groups will thus be an interesting object of study from the point of view of computable analysis alone, providing a natural space on which to test results that could apply to spaces that are not effectively Polish.

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A topological zero-one law and elementary equivalence of finitely generated groups

Cited by 9 publications

References 50 publications

Quasi‐isometric diversity of marked groups

Quasi‐isometric diversity of marked groups

Rich groups, weak second order logic, and applications

Computable analysis on the space of marked groups

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