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A geometric characterization of free formations of profinite groups. (English. Russian original) Zbl 0725.20021

Sib. Math. J. 30, No. 2, 227-235 (1989); translation from Sib. Mat. Zh. 30, No. 2(174), 73-84 (1989).
The paper presents a characterization of free formations of profinite groups in terms of profinite graphs. A theory of covering spaces of profinite graphs is constructed. Furthermore the concepts of profinite tree and simply connected graph are compared. It is shown that any projective group is the fundamental group of some graph and, finally, some characterizations of free pro-\(C\)-groups over a pointed profinite space in terms of the action of these groups on connected simply connected graphs are given.

MSC:

20E18 Limits, profinite groups
05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20F17 Formations of groups, Fitting classes
05C10 Planar graphs; geometric and topological aspects of graph theory
05C05 Trees
57S25 Groups acting on specific manifolds
20E08 Groups acting on trees
20F34 Fundamental groups and their automorphisms (group-theoretic aspects)
57M05 Fundamental group, presentations, free differential calculus
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References:

[1] D. Gildenhuys and L. Ribes, ?Profinite groups and Boolean graphs,? J. Pure Appl. Algebra,12, 21-47 (1978). · Zbl 0428.20018
[2] P. A. Zalesskii and O. V. Mel’nikov, ?Subgroups of profinite groups acting on trees,? Preprint, Akad. Nauk BSSR, Inst. Mat., No. 32, Minsk (1986).
[3] N. Bourbaki, General Topology [Russian translation], Nauka, Moscow (1975).
[4] D. Gildenhuys and C.-K. Lim, ?Free pro-C-groups,? Math. Z.,125, 233-254 (1972). · Zbl 0221.20048
[5] A. Brumer, ?Pseudo-compact algebras, profinite groups and class formations,? J. Algebra,4, 442-470 (1966). · Zbl 0146.04702
[6] L. Ribes, ?On amalgamated products of profinite groups,? Math. Z.,123, No. 4, 357-364 (1971). · Zbl 0218.20031
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