On The Profinite Topology on a Free Group

Ribes, Luis; Zalesskiĭ, Pavel

doi:10.1112/blms/25.1.37

Cited by 123 publications

(97 citation statements)

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“…In the end, the Type II conjecture was proved independently (and at virtually the same time) by Ash [8] and Ribes and Zalesskiȋ [39]. Ash's proof used inverse monoids, automata, and geometric arguments involving Cayley graphs of finite groups to obtain the result directly.…”

Section: Introductionmentioning

confidence: 97%

The geometry of profinite graphs with applications to free groups and finite monoids

Auinger

Steinberg

2003

Trans. Amer. Math. Soc.

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Abstract. We initiate the study of the class of profinite graphs Γ defined by the following geometric property: for any two vertices v and w of Γ, there is a (unique) smallest connected profinite subgraph of Γ containing them; such graphs are called tree-like. Profinite trees in the sense of Gildenhuys and Ribes are tree-like, but the converse is not true. A profinite group is then said to be dendral if it has a tree-like Cayley graph with respect to some generating set; a Bass-Serre type characterization of dendral groups is provided. Also, such groups (including free profinite groups) are shown to enjoy a certain small cancellation condition.We define a pseudovariety of groups H to be arboreous if all finitely generated free pro-H groups are dendral (with respect to a free generating set). Our motivation for studying such pseudovarieties of groups is to answer several open questions in the theory of profinite topologies and the theory of finite monoids. We prove, for arboreous pseudovarieties H, a pro-H analog of the Ribes and Zalesskiȋ product theorem for the profinite topology on a free group. Also, arboreous pseudovarieties are characterized as precisely the solutions H to the much studied pseudovariety equation J * H = J m H.

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Section: Introductionmentioning

confidence: 97%

The geometry of profinite graphs with applications to free groups and finite monoids

Auinger

Steinberg

2003

Trans. Amer. Math. Soc.

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“…In an equivalent form [10,11], Ash's theorem was rediscovered by Herwig and Lascar [25]. The stronger conjecture, equivalent to the Rhodes type II conjecture [23], was also proved by Ribes and Zalesskiȋ [34]. An elementary and constructive proof was obtained recently by Auinger [18] (see also [19]).…”

Section: Introductionmentioning

confidence: 92%

“…This terminology is justified by the fact that the above formulas hold for the pseudovariety G of all finite groups, where the closures are computed in the free group [31, 32,34].…”

Section: The Pin-reutenauer Proceduresmentioning

confidence: 99%

Closures of regular languages for profinite topologies

2014

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Abstract. The Pin-Reutenauer algorithm gives a method, that can be viewed as a descriptive procedure, to compute the closure in the free group of a regular language with respect to the Hall topology.A similar descriptive procedure is shown to hold for the pseudovariety A of aperiodic semigroups, where the closure is taken in the free aperiodic ω-semigroup. It is inherited by a subpseudovariety of a given pseudovariety if both of them enjoy the property of being full. The pseudovariety A, as well as some of its subpseudovarieties are shown to be full.The interest in such descriptions stems from the fact that, for each of the main pseudovarieties V in our examples, the closures of two regular languages are disjoint if and only if the languages can be separated by a language whose syntactic semigroup lies in V. In the cases of A and of the pseudovariety DA of semigroups in which all regular elements are idempotents, this is a new result.

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“…It survived as a conjecture almost twenty years and became a theorem after independent and deep work of Ash [2] and Ribes and Zalesskiȋ [17]. For history, motivation and consequences of the conjecture we refer the reader to [12].…”

Section: Introductionmentioning

confidence: 99%