Computable topological abelian groups

Lupini, Martino; Melnikov, A. V.; Nies, André

doi:10.48550/arxiv.2105.12897

Cited by 3 publications

(15 citation statements)

References 34 publications

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“…groups that are neither discrete nor compact. The work [22] states that in the abelian case, the notion of computable presentability given in the present paper is equivalent to these notions. We prove this in Section 8 of the present paper.…”

Section: Finite Groups and Epimorphisms ψmentioning

confidence: 93%

“…Our work [22] with Lupini focusses on abelian locally compact groups. We introduce two notions of computable presentation for abelian t.d.l.c.…”

Section: Finite Groups and Epimorphisms ψmentioning

confidence: 99%

“…Computability for particular subclasses of t.d.l.c. groups Our recent work [22], joint with Lupini, studies locally compact abelian groups with computable presentations. In particular we investigate the algorithmic content of Pontryagin-van Kampen duality for such groups.…”

Section: Function [T ]×[T ] → [T ] and The Function Det : [T ] →mentioning

confidence: 99%

“…group is procountable, i.e., an inverse limit of countable groups. This enabled us in [22] to use a notion of computable presentation for t.d.l.c. abelian groups convenient for the given context.…”

Section: Function [T ]×[T ] → [T ] and The Function Det : [T ] →mentioning

confidence: 99%

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Computably totally disconnected locally compact groups

Melnikov¹,

Nies²

2022

Preprint

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We study totally disconnected, locally compact (t.d.l.c.) groups from an algorithmic perspective. We give various approaches to defining computable presentations of t.d.l.c. groups, and show their equivalence. In the process, we obtain an algorithmic Stone-type duality between t.d.l.c. groups and certain countable ordered groupoids given by the compact open cosets. We exploit the flexibility given by these different approaches to show that several natural groups, such as Aut(T d ) and SLn(Qp), have computable presentations. We show that many construction leading from t.d.l.c. groups to new t.d.l.c. groups have algorithmic versions that stay within the class of computably presented t.d.l.c. groups. This leads to further examples, such as PGLn(Qp). We study whether objects associated with computably t.d.l.c. groups are computable: the modular function, the scale function, and Cayley-Abels graphs in the compactly generated case. We give a criterion when computable presentations of t.d.l.c. groups are unique up to computable isomorphism, and apply it to Qp as an additive group, and the semidirect product Z ⋉ Qp.

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Section: Finite Groups and Epimorphisms ψmentioning

confidence: 93%

“…Our work [22] with Lupini focusses on abelian locally compact groups. We introduce two notions of computable presentation for abelian t.d.l.c.…”

Section: Finite Groups and Epimorphisms ψmentioning

confidence: 99%

Section: Function [T ]×[T ] → [T ] and The Function Det : [T ] →mentioning

confidence: 99%

Section: Function [T ]×[T ] → [T ] and The Function Det : [T ] →mentioning

confidence: 99%

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Computably totally disconnected locally compact groups

Melnikov¹,

Nies²

2022

Preprint

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“…A compact computable Polish space is effectively compact if there is a (partial) Turing functional that given a countable cover of the space outputs it finite subcover (and is undefined otherwise). This is equivalent to saying that, for every n, we can uniformly produce at least one finite open 2 −n -cover of the space by basic open balls; see [11,Remark 2.5]. The following elementary fact is well-known: Lemma 4.8.…”

Section: Pseudofinite Groups Cmentioning

confidence: 99%

Logic Blog 2021

Nies¹

2022

Preprint

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ContentsPart 1. Group theory and its connections to logic 1. Berdinsky and Nies: Abelian groups of rank 2 2. Lupini and Nies: Extensions of abelian groups 3. Nies: questions on groups and logic Part 2. Metric spaces and descriptive set theory 4. Melnikov: Polish metric spaces and t.d.l.c. groups Part 3. FA-presentable structures 5. IMS workshop, problem session on FA-presentable structures Part 4. Set theory 6. Yu: Some consequences of Turing determinacy and strong Turing determinacy 7. Yu: On the Hausdorff dimension of Hamel bases. Part 5. Mathematical logic and quantum mechanics 8. Nies: The spectral gap problem for spin chains References

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New degree spectra of Polish spaces

Melnikov¹

2021

Sib. Mat. Zh.

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Computable topological abelian groups

Cited by 3 publications

References 34 publications

Computably totally disconnected locally compact groups

Computably totally disconnected locally compact groups

Logic Blog 2021

New degree spectra of Polish spaces

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