Profinite Graphs and Groups

Ribes, Luis

doi:10.1007/978-3-319-61199-0

Cited by 47 publications

(147 citation statements)

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“…This agrees with the standard terminology used in the theory of pro‐finite groups (cf. .) Observe that every Sylow subgroup of G is closed.…”

Section: Terminology and Basic Factsmentioning

confidence: 99%

Abelian pro-countable groups and non-Borel orbit equivalence relations

Malicki

2016

Math. Log. Quart.

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We study topological groups that can be defined as Polish, pro‐countable abelian groups, as non‐archimedean abelian groups or as quasi‐countable abelian groups, i.e., Polish subdirect products of countable, discrete groups, endowed with the product topology. We characterize tame groups in this class, i.e., groups all of whose continuous actions on a Polish space induce a Borel orbit equivalence relation, and relatively tame groups, i.e., groups all of whose diagonal actions α×β induce a Borel orbit equivalence relation, provided that α,β are continuous actions inducing Borel orbit equivalence relations.

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“…This agrees with the standard terminology used in the theory of pro‐finite groups (cf. .) Observe that every Sylow subgroup of G is closed.…”

Section: Terminology and Basic Factsmentioning

confidence: 99%

Abelian pro-countable groups and non-Borel orbit equivalence relations

Malicki

2016

Math. Log. Quart.

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“…Since

H / N \in C

and

scriptC

consists of finite groups by the assumption, we can deduce that

| H : N | < \infty

, so that

X

is a finite union of cosets modulo

N

. Consequently,

X

scriptC

‐closed in

G

as a finite union of

scriptC

‐closed sets, because each coset

N h_{i}

scriptC

‐closed in

G

. (b)See the proof of [, Lemma 3.1.4(a)].

□

…”

Section: The Pro‐scriptc Topology On Subdirect Productsmentioning

confidence: 93%

On conjugacy separability of fibre products

Minasyan

2017

Proc. London Math. Soc.

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In this paper we study conjugacy separability of subdirect products of two free (or hyperbolic) groups. We establish necessary and sufficient criteria and apply them to fibre products to produce a finitely presented group $G_1$ in which all finite index subgroups are conjugacy separable, but which has an index $2$ overgroup that is not conjugacy separable. Conversely, we construct a finitely presented group $G_2$ which has a non-conjugacy separable subgroup of index $2$ such that every finite index normal overgroup of $G_2$ is conjugacy separable. The normality of the overgroup is essential in the last example, as such a group $G_2$ will always posses an index $3$ overgroup that is not conjugacy separable. Finally, we characterize $p$-conjugacy separable subdirect products of two free groups, where $p$ is a prime. We show that fibre products provide a natural correspondence between residually finite $p$-groups and $p$-conjugacy separable subdirect products of two non-abelian free groups. As a consequence, we deduce that the open question about the existence of an infinite finitely presented residually finite $p$-group is equivalent to the question about the existence of a finitely generated $p$-conjugacy separable full subdirect product of infinite index in the direct product of two free groups.Comment: v2: 38 pages; this is the version accepted for publicatio

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“…Then there is a unique uniformly continuous map g : Y → X such that φ i g = g i for all i ∈ I. The map g exists and is unique by the general theory of inverse limits of sets, and it is uniformly continuous by Corollary 3.9 [9].…”

Section: Inverse Limits Of Cofinite Spacesmentioning

confidence: 96%

“…We say that (X i , φ ij ) is an inverse system of uniform spaces if (i) each X i is a uniform space, and (ii) for all i ≤ j, φ ij : X j → X i is uniformly continuous. The set X = lim ← − X i endowed with the coarsest uniformity for which the canonical maps φ i : X → X i are uniformly continuous is called the inverse limit of the inverse system of uniform spaces [9].…”

Section: Inverse Limits Of Cofinite Spacesmentioning

confidence: 99%

Cofinite graphs and their profinite completions

Acharyya

Corson

Das

2017

ejgta

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We generalize the idea of cofinite groups, due to B. Hartley, [2]. First we define cofinite spaces in general. Then, as a special situation, we study cofinite graphs and their uniform completions.The idea of constructing a cofinite graph starts with defining a uniform topological graph Γ, in an appropriate fashion. We endow abstract graphs with uniformities corresponding to separating filter bases of equivalence relations with finitely many equivalence classes over Γ. It is established that for any cofinite graph there exists a unique cofinite completion.

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Profinite Graphs and Groups

Cited by 47 publications

References 0 publications

Abelian pro-countable groups and non-Borel orbit equivalence relations

Abelian pro-countable groups and non-Borel orbit equivalence relations

On conjugacy separability of fibre products

Cofinite graphs and their profinite completions

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